Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Lecture 5 – Limits at Infinity and Asymptotes
Limits at Infinity
Horizontal Asymptotes
Limits at Infinity for Polynomials
Limit of a Reciprocal Power
The End Behavior of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
Example 17 – Limit of Rational Function
Example 18 – More Rational Functions
Limits at Infinity for the Exponential Function
Example 19 – Limits of Exponential Functions
Function Dominance
Example 20 – Exponential and Power Functions
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Horizontal Asymptotes and Limits at Infinity
Example 21 – Investigating Asymptotes
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
1/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity
Limits at Infinity
If a function f has a domain that is unbounded, that is, one of the
endpoints of its domain is ±∞, we can determine the long term
behavior of the function using a limit at infinity.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
2/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity
Limits at Infinity
If a function f has a domain that is unbounded, that is, one of the
endpoints of its domain is ±∞, we can determine the long term
behavior of the function using a limit at infinity.
Definition (Limits at Infinity)
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
2/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity
Limits at Infinity
If a function f has a domain that is unbounded, that is, one of the
endpoints of its domain is ±∞, we can determine the long term
behavior of the function using a limit at infinity.
Definition (Limits at Infinity)
We say that
lim f (x) = L
x→ ∞
if f (x) can be made arbitrarily close to L by making x sufficiently
large and positive
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
2/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity
Limits at Infinity
If a function f has a domain that is unbounded, that is, one of the
endpoints of its domain is ±∞, we can determine the long term
behavior of the function using a limit at infinity.
Definition (Limits at Infinity)
We say that
lim f (x) = L
x→ ∞
if f (x) can be made arbitrarily close to L by making x sufficiently
large and positive, and we say that
lim f (x) = M
x→−∞
if f (x) can be made arbitrarily close to M by making x sufficiently
large and negative.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
2/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
If
lim f (x) = L
x→ ∞
then the graph of f has a horizontal asymptote along the line y = L
on the right
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
If
lim f (x) = L
x→ ∞
then the graph of f has a horizontal asymptote along the line y = L
on the right, and if
lim f (x) = M
x→−∞
then the graph of f has a horizontal asymptote along the line y = M
on the left.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
If
lim f (x) = L
x→ ∞
then the graph of f has a horizontal asymptote along the line y = L
on the right, and if
lim f (x) = M
x→−∞
then the graph of f has a horizontal asymptote along the line y = M
on the left.
If f has an unbounded domain, but f (x) does not approach a finite
value as x goes to ±∞, there are two possibilities:
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
If
lim f (x) = L
x→ ∞
then the graph of f has a horizontal asymptote along the line y = L
on the right, and if
lim f (x) = M
x→−∞
then the graph of f has a horizontal asymptote along the line y = M
on the left.
If f has an unbounded domain, but f (x) does not approach a finite
value as x goes to ±∞, there are two possibilities:
the limit at infinity is infinite
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
If
lim f (x) = L
x→ ∞
then the graph of f has a horizontal asymptote along the line y = L
on the right, and if
lim f (x) = M
x→−∞
then the graph of f has a horizontal asymptote along the line y = M
on the left.
If f has an unbounded domain, but f (x) does not approach a finite
value as x goes to ±∞, there are two possibilities:
the limit at infinity is infinite
the limit at infinity does not exist
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
Horizontal Asymptotes
If
lim f (x) = L
x→ ∞
then the graph of f has a horizontal asymptote along the line y = L
on the right, and if
lim f (x) = M
x→−∞
then the graph of f has a horizontal asymptote along the line y = M
on the left.
If f has an unbounded domain, but f (x) does not approach a finite
value as x goes to ±∞, there are two possibilities:
the limit at infinity is infinite
the limit at infinity does not exist
In either case, the graph of f does not have a horizontal asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
3/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limit of a Reciprocal Power
Limit of a Reciprocal Power
Limit of a Reciprocal Power
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
4/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limit of a Reciprocal Power
Limit of a Reciprocal Power
Limit of a Reciprocal Power
If r > 0, then
lim
x→ ∞
Clint Lee
1
=0
xr
Math 112 Lecture 5: Limits at Infinity and Asymptotes
4/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limit of a Reciprocal Power
Limit of a Reciprocal Power
Limit of a Reciprocal Power
If r > 0, then
lim
x→ ∞
and
lim
x→−∞
Clint Lee
1
=0
xr
1
=0
xr
Math 112 Lecture 5: Limits at Infinity and Asymptotes
4/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limit of a Reciprocal Power
Limit of a Reciprocal Power
Limit of a Reciprocal Power
If r > 0, then
lim
x→ ∞
and
lim
x→−∞
1
=0
xr
1
=0
xr
if r is a positive integer or is a fraction that results is an even root.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
4/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
The End Behavior of a Polynomial
The End Behavior of a Polynomial
Limits at Infinity for Polynomials
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
5/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
The End Behavior of a Polynomial
The End Behavior of a Polynomial
Limits at Infinity for Polynomials
For the nth degree polynomial function
p(x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
the the end behavior is given by the limit at infinity
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
5/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
The End Behavior of a Polynomial
The End Behavior of a Polynomial
Limits at Infinity for Polynomials
For the nth degree polynomial function
p(x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
the the end behavior is given by the limit at infinity
lim p(x)
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
5/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
The End Behavior of a Polynomial
The End Behavior of a Polynomial
Limits at Infinity for Polynomials
For the nth degree polynomial function
p(x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
the the end behavior is given by the limit at infinity
lim p(x)
x→±∞
which is always infinite and whose sign depends only on the sign of
an and on whether n even or odd.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
5/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
The at infinity of a polynomial is
lim p(x) = lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
x→±∞
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
6/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
The at infinity of a polynomial is
lim p(x) = lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
x→±∞
x→±∞
a2
a1
a a
+ n−
+ 0n
= lim xn an + n−1 + · · · + n−
2
1
x→±∞
x
x
x
x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
6/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
The at infinity of a polynomial is
lim p(x) = lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
x→±∞
x→±∞
a2
a1
a a
+ n−
+ 0n
= lim xn an + n−1 + · · · + n−
2
1
x→±∞
x
x
x
x
Using the the limit at infinity of a reciprocal power, gives
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
6/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
The at infinity of a polynomial is
lim p(x) = lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
x→±∞
x→±∞
a2
a1
a a
+ n−
+ 0n
= lim xn an + n−1 + · · · + n−
2
1
x→±∞
x
x
x
x
Using the the limit at infinity of a reciprocal power, gives


a2
an−1
a1
a0
+ · · · + n−2 + n−1 + n 
an +
x}
x
lim p(x) = lim xn 
| {z
|x {z } |x {z } |{z}
x→±∞
x→±∞
−→0
Clint Lee
−→0
−→0
−→0
Math 112 Lecture 5: Limits at Infinity and Asymptotes
6/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
The at infinity of a polynomial is
lim p(x) = lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
x→±∞
x→±∞
a2
a1
a a
+ n−
+ 0n
= lim xn an + n−1 + · · · + n−
2
1
x→±∞
x
x
x
x
Using the the limit at infinity of a reciprocal power, gives


a2
an−1
a1
a0
+ · · · + n−2 + n−1 + n 
an +
x}
x
lim p(x) = lim xn 
| {z
|x {z } |x {z } |{z}
x→±∞
x→±∞
−→0
−→0
−→0
−→0
All terms except the first go to zero.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
6/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
The at infinity of a polynomial is
lim p(x) = lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0
x→±∞
x→±∞
a2
a1
a a
+ n−
+ 0n
= lim xn an + n−1 + · · · + n−
2
1
x→±∞
x
x
x
x
Using the the limit at infinity of a reciprocal power, gives


a2
an−1
a1
a0
+ · · · + n−2 + n−1 + n 
an +
x}
x
lim p(x) = lim xn 
| {z
|x {z } |x {z } |{z}
x→±∞
x→±∞
−→0
−→0
−→0
−→0
All terms except the first go to zero. Thus,
lim p(x) = an lim xn
x→±∞
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
6/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
Now, if n > 0, then
n even ⇒ lim xn = ∞
x→±∞
and
Clint Lee
n odd ⇒ lim xn = ±∞
x→±∞
Math 112 Lecture 5: Limits at Infinity and Asymptotes
7/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
Now, if n > 0, then
n even ⇒ lim xn = ∞
x→±∞
and
n odd ⇒ lim xn = ±∞
x→±∞
Thus,
lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 =
x→ ∞
Clint Lee
(
+∞ if an > 0
−∞ if an < 0
Math 112 Lecture 5: Limits at Infinity and Asymptotes
7/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Evaluating the Limit at Infinity of a Polynomial
Evaluating the Limit at Infinity of a Polynomial
Now, if n > 0, then
n even ⇒ lim xn = ∞
x→±∞
and
n odd ⇒ lim xn = ±∞
x→±∞
Thus,
lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 =
x→ ∞
(
+∞ if an > 0
−∞ if an < 0
and
lim an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 =
(
+∞ if n even & an > 0 or n odd & an < 0
−∞ if n even & an < 0 or n odd & an > 0
x→−∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
7/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Example 17 – Limit of Rational Function
Evaluate the limit
lim
x→ ∞
4x3 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
8/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17
To make use of the limits just discussed, the expression must involve
terms with reciprocal powers. Thus divide the numerator and
denominator by the highest power of x in the denominator, here x3 .
This gives
3
4− +
4x3 − 3x2 + 2x − 4
x
lim
= lim
2
x→∞ 5x3 + 2x2 − 4x + 3
x→ ∞
5+ −
x
Clint Lee
2
4
− 3
x2
x
3
4
+ 3
x2
x
Math 112 Lecture 5: Limits at Infinity and Asymptotes
9/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17
To make use of the limits just discussed, the expression must involve
terms with reciprocal powers. Thus divide the numerator and
denominator by the highest power of x in the denominator, here x3 .
This gives
2
3
4
4− + 2 − 3
4x3 − 3x2 + 2x − 4
x
x
x
lim
= lim
2
3
4
x→∞ 5x3 + 2x2 − 4x + 3
x→ ∞
5+ − 2 + 3
x x
x
3
2
4
4−
+ 2 − 3
x
x
x
|{z} |{z}
|{z}
= lim
x→ ∞
−→0
−→0
Clint Lee
−→0
−→0
2
3
4
5+
− 2 + 3
x
x
x
|{z} |{z} |{z}
−→0
−→0
Math 112 Lecture 5: Limits at Infinity and Asymptotes
9/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17
To make use of the limits just discussed, the expression must involve
terms with reciprocal powers. Thus divide the numerator and
denominator by the highest power of x in the denominator, here x3 .
This gives
2
3
4
4− + 2 − 3
4x3 − 3x2 + 2x − 4
x
x
x
lim
= lim
2
3
4
x→∞ 5x3 + 2x2 − 4x + 3
x→ ∞
5+ − 2 + 3
x x
x
3
2
4
4−
+ 2 − 3
x
x
x
|{z} |{z}
|{z}
= lim
x→ ∞
−→0
−→0
4
=
5
Clint Lee
−→0
−→0
2
3
4
5+
− 2 + 3
x
x
x
|{z} |{z} |{z}
−→0
−→0
Math 112 Lecture 5: Limits at Infinity and Asymptotes
9/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17 continue
An identical argument for x → −∞ shows that
lim
x→−∞
4
4x3 − 3x2 + 2x − 4
=
3
2
5
5x + 2x − 4x + 3
Hence the graph of the rational function
r(x) =
4x3 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
has a horizontal asymptote along the
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
10/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17 continue
An identical argument for x → −∞ shows that
lim
x→−∞
4
4x3 − 3x2 + 2x − 4
=
3
2
5
5x + 2x − 4x + 3
Hence the graph of the rational function
r(x) =
4x3 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
has a horizontal asymptote along the line y =
the left.
Clint Lee
4
on both the right and
5
Math 112 Lecture 5: Limits at Infinity and Asymptotes
10/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17 continue
An identical argument for x → −∞ shows that
lim
x→−∞
4
4x3 − 3x2 + 2x − 4
=
3
2
5
5x + 2x − 4x + 3
Hence the graph of the rational function
r(x) =
4x3 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
has a horizontal asymptote along the line y =
the left.
4
on both the right and
5
Horizontal Asymptotes of a Rational Function
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
10/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17 continue
An identical argument for x → −∞ shows that
lim
x→−∞
4
4x3 − 3x2 + 2x − 4
=
3
2
5
5x + 2x − 4x + 3
Hence the graph of the rational function
r(x) =
4x3 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
has a horizontal asymptote along the line y =
the left.
4
on both the right and
5
Horizontal Asymptotes of a Rational Function
In general, for a rational function in which the degrees of polynomials
in the numerator and denominator are equal, the limit at ±∞ equals
the ratio of the coefficients of the highest powers, call it R,
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
10/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 17 – Limit of Rational Function
Solution: Example 17 continue
An identical argument for x → −∞ shows that
lim
x→−∞
4
4x3 − 3x2 + 2x − 4
=
3
2
5
5x + 2x − 4x + 3
Hence the graph of the rational function
r(x) =
4x3 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
has a horizontal asymptote along the line y =
the left.
4
on both the right and
5
Horizontal Asymptotes of a Rational Function
In general, for a rational function in which the degrees of polynomials
in the numerator and denominator are equal, the limit at ±∞ equals
the ratio of the coefficients of the highest powers, call it R, and the
function has a horizontal asymptote along a line y = R on both sides.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
10/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
Clint Lee
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Solution:
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Solution:
2
4
4
3
− 2+ 3− 4
4x3 − 3x2 + 2x − 4
x
x
x
x
= lim
(a) lim
4
3
2
x→±∞
x→±∞ 5x4 + 2x2 − 4x + 3
5+ 2 − 3 + 4
x
x
x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Solution:
2
4
4
3
− 2+ 3− 4
4x3 − 3x2 + 2x − 4
x
x
x
x
=0
= lim
(a) lim
4
3
2
x→±∞
x→±∞ 5x4 + 2x2 − 4x + 3
5+ 2 − 3 + 4
x
x
x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Solution:
2
4
4
3
− 2+ 3− 4
4x3 − 3x2 + 2x − 4
x
x
x
x
=0
= lim
(a) lim
4
3
2
x→±∞
x→±∞ 5x4 + 2x2 − 4x + 3
5+ 2 − 3 + 4
x
x
x
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Solution:
2
4
4
3
− 2+ 3− 4
4x3 − 3x2 + 2x − 4
x
x
x
x
=0
= lim
(a) lim
4
3
2
x→±∞
x→±∞ 5x4 + 2x2 − 4x + 3
5+ 2 − 3 + 4
x
x
x
4
2
3
4x − + 2 − 3
4x4 − 3x2 + 2x − 4
x
x
x
(b) lim
= lim
4
3
2
x→±∞ 5x3 + 2x2 − 4x + 3
x→±∞
5+ − 2 + 3
x x
x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 18 – More Rational Functions
Example 18 – More Rational Functions
Evaluate the limits
(a)
lim
x→±∞
4x3 − 3x2 + 2x − 4
5x4 + 2x2 − 4x + 3
(b)
lim
x→±∞
4x4 − 3x2 + 2x − 4
5x3 + 2x2 − 4x + 3
Solution:
2
4
4
3
− 2+ 3− 4
4x3 − 3x2 + 2x − 4
x
x
x
x
=0
= lim
(a) lim
4
3
2
x→±∞
x→±∞ 5x4 + 2x2 − 4x + 3
5+ 2 − 3 + 4
x
x
x
4
2
3
4x − + 2 − 3
4x4 − 3x2 + 2x − 4
x
x
x
(b) lim
= lim
= ±∞
4
3
2
x→±∞ 5x3 + 2x2 − 4x + 3
x→±∞
5+ − 2 + 3
x x
x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
11/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
The graph of the natural exponential
function f (x) = ex looks like this.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
12/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
The graph of the natural exponential
function f (x) = ex looks like this.
f (x) = ex
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
12/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
The graph of the natural exponential
function f (x) = ex looks like this. The
graph of the function g(x) = e−x is
obtained by reflecting across the y-axis.
f (x) = ex
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
12/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
The graph of the natural exponential
function f (x) = ex looks like this. The
graph of the function g(x) = e−x is
obtained by reflecting across the y-axis. It
looks like this.
f (x) = ex
g( x ) = e− x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
12/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
From the graphs just shown
lim ex = ∞
x→ ∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
From the graphs just shown
lim ex = ∞
x→ ∞
Clint Lee
and
lim ex = 0
x→−∞
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
From the graphs just shown
lim ex = ∞
x→ ∞
and
lim ex = 0
x→−∞
and
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
From the graphs just shown
lim ex = ∞
x→ ∞
and
lim ex = 0
x→−∞
and
lim e−x
x→−∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
From the graphs just shown
lim ex = ∞
x→ ∞
and
lim ex = 0
x→−∞
and
lim e−x = ∞
x→−∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Limits at Infinity for the Exponential Function
Limits at Infinity for the Exponential Function
From the graphs just shown
lim ex = ∞
and
lim e−x = ∞
and
x→ ∞
lim ex = 0
x→−∞
and
x→−∞
Clint Lee
lim e−x = 0
x→ ∞
Math 112 Lecture 5: Limits at Infinity and Asymptotes
13/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
Clint Lee
lim
x→ ∞
2ex + 1
ex + 2
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
ex + 2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
=
ex + 2
0+2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
1
=
=
ex + 2
0+2
2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
1
=
=
ex + 2
0+2
2
(b) Now we use lim e−x = 0. To do the we first multiply
x→ ∞
numerator and denominator by e−x to give
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
1
=
=
ex + 2
0+2
2
(b) Now we use lim e−x = 0. To do the we first multiply
x→ ∞
numerator and denominator by e−x to give
lim
x→ ∞
2ex + 1
ex + 2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
1
=
=
ex + 2
0+2
2
(b) Now we use lim e−x = 0. To do the we first multiply
x→ ∞
numerator and denominator by e−x to give
−x x
e
2e + 1
2ex + 1
= lim
lim
x→ ∞
x → ∞ ex + 2
ex + 2
e−x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
1
=
=
ex + 2
0+2
2
(b) Now we use lim e−x = 0. To do the we first multiply
x→ ∞
numerator and denominator by e−x to give
−x x
e
2 + e−x
2e + 1
2ex + 1
= lim
=
lim
lim x
x→∞ 1 + 2e−x
x→ ∞
x→ ∞ e + 2
ex + 2
e−x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Example 19 – Limits of Exponential Functions
Evaluate the limits
2ex + 1
(a)
lim
x→−∞ ex + 2
(b)
lim
x→ ∞
2ex + 1
ex + 2
Solution:
(a) Since lim ex = 0 we have
x→−∞
lim
x→−∞
2ex + 1
2·0+1
1
=
=
ex + 2
0+2
2
(b) Now we use lim e−x = 0. To do the we first multiply
x→ ∞
numerator and denominator by e−x to give
−x x
e
2 + e−x
2e + 1
2ex + 1
= lim
=
lim
=2
lim x
x→∞ 1 + 2e−x
x→ ∞
x→ ∞ e + 2
ex + 2
e−x
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
14/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Solution: Example 19 continued
2ex + 1
has a horizontal asymptotes
ex + 2
on the left and y = 2 on the right. Its graph looks like
The graph of the function f (x) =
along y =
this.
1
2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
15/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Solution: Example 19 continued
2ex + 1
has a horizontal asymptotes
ex + 2
on the left and y = 2 on the right. Its graph looks like
The graph of the function f (x) =
along y =
this.
1
2
2
f (x) =
2e x + 1
ex + 2
1
2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
15/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Horizontal Asymptotes for Functions Involving the Exponential Function
Horizontal Asymptotes for Exponential Functions
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
16/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Horizontal Asymptotes for Functions Involving the Exponential Function
Horizontal Asymptotes for Exponential Functions
For a function involving an exponential function the limits at +∞ and
−∞ can be different.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
16/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 19 – Limits of Exponential Functions
Horizontal Asymptotes for Functions Involving the Exponential Function
Horizontal Asymptotes for Exponential Functions
For a function involving an exponential function the limits at +∞ and
−∞ can be different. This means that such functions can have a
different horizontal asymptote on the right (x → ∞) and on the left
(x → −∞).
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
16/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Function Dominance
When comparing functions, an important question is: “which
function grows faster as x gets large?” This question can be answered
using limits at infinity.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
17/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Function Dominance
When comparing functions, an important question is: “which
function grows faster as x gets large?” This question can be answered
using limits at infinity.
Function Dominance
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
17/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Function Dominance
When comparing functions, an important question is: “which
function grows faster as x gets large?” This question can be answered
using limits at infinity.
Function Dominance
For two function f and g, if
lim
x→ ∞
f (x)
=0
g(x)
we say that g dominates f as x gets large and positive.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
17/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Function Dominance
When comparing functions, an important question is: “which
function grows faster as x gets large?” This question can be answered
using limits at infinity.
Function Dominance
For two function f and g, if
lim
x→ ∞
f (x)
=0
g(x)
we say that g dominates f as x gets large and positive.
Equivalently, we have
g(x)
lim
=∞
x→ ∞ f (x)
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
17/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Example 20 – Exponential and Power Functions
Consider the family of functions
f (x) = xn e−x =
x
ex
for x ≥ 0
where the parameter n is a positive
integer.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
18/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Example 20 – Exponential and Power Functions
Consider the family of functions
f (x) = xn e−x =
x
ex
for x ≥ 0
where the parameter n is a positive
integer.
The graph of the single member of the
family with n = 1 looks like this.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
18/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Example 20 – Exponential and Power Functions
Consider the family of functions
f (x) = xn e−x =
x
ex
for x ≥ 0
where the parameter n is a positive
integer.
The graph of the two members of the
family with n = 1 and n = 2 look like
this.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
18/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Example 20 – Exponential and Power Functions
Consider the family of functions
f (x) = xn e−x =
x
ex
for x ≥ 0
where the parameter n is a positive
integer.
The graph of the three members of the
family with n = 1, n = 2, and n = 3 look
like this.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
18/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Example 20 – Exponential and Power Functions
Consider the family of functions
f (x) = xn e−x =
x
ex
for x ≥ 0
where the parameter n is a positive
integer.
The graph of the four members of the
family with n = 1, n = 2, n = 3, and
n = 4 look like this.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
18/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Example 20 – Exponential and Power Functions
Consider the family of functions
f (x) = xn e−x =
x
ex
for x ≥ 0
where the parameter n is a positive
integer.
The graph of the four members of the
family with n = 1, n = 2, n = 3, and
n = 4 look like this.
Based on these graphs what can you
conclude about the dominance
relationship between the exponential
function ex and the power function xn ?
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
18/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Solution: Example 20
From the graphs it appears that for any integer n ≥ 1 we have
lim
x→ ∞
Clint Lee
xn
=0
ex
Math 112 Lecture 5: Limits at Infinity and Asymptotes
19/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
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Example 20 – Exponential and Power Functions
Solution: Example 20
From the graphs it appears that for any integer n ≥ 1 we have
lim
x→ ∞
xn
=0
ex
since for each of the graphs the x-axis, y = 0, is a horizontal
asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
19/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Solution: Example 20
From the graphs it appears that for any integer n ≥ 1 we have
lim
x→ ∞
xn
=0
ex
since for each of the graphs the x-axis, y = 0, is a horizontal
asymptote.
Thus, we conclude that the exponential function ex dominates xn as
x → ∞.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
19/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Solution: Example 20
From the graphs it appears that for any integer n ≥ 1 we have
lim
x→ ∞
xn
=0
ex
since for each of the graphs the x-axis, y = 0, is a horizontal
asymptote.
Thus, we conclude that the exponential function ex dominates xn as
x → ∞.
In other words, the exponential function grows more rapidly than the
power function as x gets large and positive.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
19/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 20 – Exponential and Power Functions
Solution: Example 20
From the graphs it appears that for any integer n ≥ 1 we have
lim
x→ ∞
xn
=0
ex
since for each of the graphs the x-axis, y = 0, is a horizontal
asymptote.
Thus, we conclude that the exponential function ex dominates xn as
x → ∞.
In other words, the exponential function grows more rapidly than the
power function as x gets large and positive.
Further, from the graphs it appears that as n gets larger, we must go
to larger x values for the dominance to take effect.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
19/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Where Are Vertical Asymptotes?
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Where Are Vertical Asymptotes?
Where the function has an infinite limit.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Where Are Vertical Asymptotes?
Where the function has an infinite limit.
There may be no vertical asymptotes, or many.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Where Are Vertical Asymptotes?
Where the function has an infinite limit.
There may be no vertical asymptotes, or many.
A rational function has a vertical asymptote for any x value
where the denominator is zero but the numerator is non-zero.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Where Are Vertical Asymptotes?
Where the function has an infinite limit.
There may be no vertical asymptotes, or many.
A rational function has a vertical asymptote for any x value
where the denominator is zero but the numerator is non-zero.
Some functions have vertical asymptotes without having an
obvious denominator to make equal to zero.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Vertical Asymptotes and Infinite Limits
Vertical Asymptotes and Infinite Limits
Where Are Vertical Asymptotes?
Where the function has an infinite limit.
There may be no vertical asymptotes, or many.
A rational function has a vertical asymptote for any x value
where the denominator is zero but the numerator is non-zero.
Some functions have vertical asymptotes without having an
obvious denominator to make equal to zero.
For example, tan x has vertical asymptotes at any odd multiple of
π
2.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
20/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
There can be at most two different horizontal asymptotes: on the
right (x → ∞) and on the left (x → −∞).
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
There can be at most two different horizontal asymptotes: on the
right (x → ∞) and on the left (x → −∞).
A rational function can have only one horizontal asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
There can be at most two different horizontal asymptotes: on the
right (x → ∞) and on the left (x → −∞).
A rational function can have only one horizontal asymptote.
If the degree of the denominator and numerator are equal, there is
a non-zero horizontal asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
There can be at most two different horizontal asymptotes: on the
right (x → ∞) and on the left (x → −∞).
A rational function can have only one horizontal asymptote.
If the degree of the denominator and numerator are equal, there is
a non-zero horizontal asymptote.
If the degree of the denominator is greater than the degree of the
numerator, there is a horizontal asymptote at y = 0.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
There can be at most two different horizontal asymptotes: on the
right (x → ∞) and on the left (x → −∞).
A rational function can have only one horizontal asymptote.
If the degree of the denominator and numerator are equal, there is
a non-zero horizontal asymptote.
If the degree of the denominator is greater than the degree of the
numerator, there is a horizontal asymptote at y = 0.
If the degree of the denominator is less than the degree of the
numerator, there is no horizontal asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Horizontal Asymptotes and Limits at Infinity
Horizontal Asymptotes and Limits at Infinity
What Are the Horizontal Asymptotes?
To identify any horizontal asymptotes for a function determine
the limits at infinity.
There can be at most two different horizontal asymptotes: on the
right (x → ∞) and on the left (x → −∞).
A rational function can have only one horizontal asymptote.
If the degree of the denominator and numerator are equal, there is
a non-zero horizontal asymptote.
If the degree of the denominator is greater than the degree of the
numerator, there is a horizontal asymptote at y = 0.
If the degree of the denominator is less than the degree of the
numerator, there is no horizontal asymptote.
Functions involving exponential functions may have different
horizontal asymptotes on the left and the right.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
21/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Example 21 – Investigating Asymptotes
For each function below, determine any vertical and horizontal
asymptotes and sketch the graph of the function. For the vertical
asymptote(s) determine the behavior of the function on either side of
the asymptote by calculating the appropriate one-sided infinite
limits.
1
x+1
x
(b) g(x) =
x+1
1
(c) h(x) =
(x + 1)2
(d) F(x) =
(a) f (x) =
(e) G(x) =
(f)
Clint Lee
H (x) =
x
(x + 1)2
x2
(x + 1)2
x3
(x + 1)2
Math 112 Lecture 5: Limits at Infinity and Asymptotes
22/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x+1 = 0
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
and
lim
x→−1+
1
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
and
lim
x→−1+
1
=∞
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
Further,
lim
x→±∞
and
lim
x→−1+
1
=∞
x+1
1
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
Further,
lim
x→±∞
and
lim
x→−1+
1
=∞
x+1
1
=0
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
and
lim
x→−1+
1
=∞
x+1
Further,
1
=0
x+1
So that, the x-axis, y = 0, is the horizontal
asymptote for the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(a)
For
1
x+1
there is a vertical asymptote where
x = −1
f (x) =
x + 1 = 0 ⇒ x = −1
The behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
= −∞
x+1
and
lim
x→−1+
1
=∞
x+1
Further,
1
=0
x+1
So that, the x-axis, y = 0, is the horizontal
asymptote for the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
23/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at
g(x) =
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
g(x) =
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
=∞
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
=∞
x+1
and
lim
x→−1+
x
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
=∞
x+1
and
lim
x→−1+
x
= −∞
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
=∞
x+1
and
Further,
lim
x→±∞
lim
x→−1+
x
= −∞
x+1
x
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
=∞
x+1
and
Further,
lim
x→±∞
lim
x→−1+
x
= −∞
x+1
x
=1
x+1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
lim
x→−1−
x
=∞
x+1
and
lim
x→−1+
x
= −∞
x+1
Further,
x
=1
x+1
So that y = 1 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(b)
For
x
x+1
again there is a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
g(x) =
x
=∞
x+1
and
lim
x→−1+
x
= −∞
x+1
x = −1
lim
x→−1−
y=1
Further,
x
=1
x+1
So that y = 1 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
24/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)2
=∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)
2
= ∞ and
lim
x→−1+
Clint Lee
1
(x + 1)2
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
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Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)
2
= ∞ and
lim
x→−1+
Clint Lee
1
(x + 1)2
=∞
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
1
h(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)
2
= ∞ and
Further,
lim
x→±∞
lim
x→−1+
1
(x + 1)2
=∞
1
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
1
h(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)
2
= ∞ and
Further,
lim
x→±∞
1
(x + 1)2
lim
x→−1+
1
(x + 1)2
=∞
=0
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
For
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)
2
= ∞ and
lim
x→−1+
1
(x + 1)2
=∞
Further,
1
=0
(x + 1)2
So that y = 0 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(c)
h(x) =
1
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1−
1
(x + 1)
2
= ∞ and
lim
x→−1+
1
(x + 1)2
x = −1
For
=∞
Further,
1
=0
(x + 1)2
So that y = 0 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
25/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
F(x) =
x
(x + 1)2
there is still a vertical asymptote at
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
F(x) =
x
(x + 1)2
there is still a vertical asymptote at x = −1.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
x
F(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
x
F(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x
(x + 1)2
= −∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
x
F(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x
(x + 1)2
Further,
lim
x→±∞
= −∞
x
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
x
F(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x
(x + 1)2
Further,
lim
x→±∞
= −∞
x
(x + 1)2
=0
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
x
F(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x
(x + 1)2
Further,
= −∞
x
=0
(x + 1)2
So that y = 0 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(d)
For
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x
(x + 1)2
Further,
x = −1
x
F(x) =
= −∞
x
=0
(x + 1)2
So that y = 0 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
26/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
G(x) =
x2
(x + 1)2
there is still a vertical asymptote at
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
G(x) =
x2
(x + 1)2
there is still a vertical asymptote at x = −1.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
x2
G(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x2
(x + 1)2
=
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
x2
G(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x2
(x + 1)2
=∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
x2
G(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
Further,
lim
x→±∞
x2
(x + 1)2
=∞
x2
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
x2
G(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
Further,
lim
x→±∞
x2
(x + 1)2
x2
(x + 1)2
=∞
=1
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
x2
G(x) =
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
Further,
x2
(x + 1)2
=∞
x2
=1
(x + 1)2
So that y = 1 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(e)
For
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
Further,
x2
(x + 1)2
x = −1
x2
G(x) =
y=1
=∞
x2
=1
(x + 1)2
So that y = 1 is the horizontal asymptote for
the graph of the function.
lim
x→±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
27/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at x = −1.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x3
(x + 1)2
=
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x3
(x + 1)2
= −∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x3
(x + 1)2
= −∞
Further,
lim
x→±∞
x3
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x3
(x + 1)2
= −∞
Further,
lim
x→±∞
x3
(x + 1)2
= ±∞
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
For
H (x) =
x3
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x3
(x + 1)2
= −∞
Further,
lim
x→±∞
x3
(x + 1)2
= ±∞
So there is no horizontal asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f)
H (x) =
x3
x = −1
For
y
=
x
−
2
(x + 1)2
there is still a vertical asymptote at x = −1.
Now the behavior on either side of the vertical
asymptote at x = −1 is given by
lim
x→−1
x3
(x + 1)2
= −∞
Further,
lim
x→±∞
x3
(x + 1)2
= ±∞
So there is no horizontal asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
28/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f) continued
Note that
x3
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
29/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f) continued
Note that
x3
(x + 1)2
=
x3 + 2x2 + x − 2x2 − 4x − 2 + 3x + 2
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
29/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f) continued
Note that
x3
(x + 1)2
=
=
x3 + 2x2 + x − 2x2 − 4x − 2 + 3x + 2
(x + 1)2
x (x + 1)2 − 2 (x + 1)2 + 3x + 2
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
29/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f) continued
Note that
x3
(x + 1)2
=
=
x3 + 2x2 + x − 2x2 − 4x − 2 + 3x + 2
(x + 1)2
x (x + 1)2 − 2 (x + 1)2 + 3x + 2
= x−2+
(x + 1)2
3x + 2
(x + 1)2
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
29/29
Limits at Infinity
Limits at Infinity for Polynomials
Limits at Infinity for the Exponential Function
Function Dominance
More on Asymptotes
Example 21 – Investigating Asymptotes
Solution: Example 21(f) continued
Note that
x3
(x + 1)2
=
=
x3 + 2x2 + x − 2x2 − 4x − 2 + 3x + 2
(x + 1)2
x (x + 1)2 − 2 (x + 1)2 + 3x + 2
= x−2+
(x + 1)2
3x + 2
(x + 1)2
Now
lim
x→±∞
3x + 2
(x + 1)2
=0
so we say that as x → ±∞ the function H (x) behaves like x − 2. We
call the line y = x − 2 a slant asymptote.
Clint Lee
Math 112 Lecture 5: Limits at Infinity and Asymptotes
29/29