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Lecture 11: Section 3.1
Evaluating Limits Algebraically
Limits at Infinity
Recall the following definition:
The function f has the limit L as x approaches
a, written
lim f (x) = L,
x→a
if the value of f (x) can be made as close to the number L as we like by taking x as close as necessary to
a, from either side, but not equal to a.
We have seen how to evaluate limits from a table of
values or graph.
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ex. Consider the graph of function f (x) given below:
Evaluate the following limits if possible:
(1) lim f (x) =
(5) lim f (x) =
(2) lim f (x) =
(6) lim f (x) =
(3) lim f (x) =
(7) lim f (x) =
x→−3
x→−2
x→−1−
(4) lim f (x) =
x→−1+
x→−1
x→2
x→4
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To Evaluate Limits Algebraically
Many limits can be found by direct substitution
using the limit properties from Lecture 10:
2 +2x−3
ex. Evaluate lim ex
x→−3
p(x)
What about a rational function R(x) =
where
q(x)
p and q are polynomials? If q(a) 6= 0, we can use
substitution: lim R(x) = R(a).
x→a
3−x
.
x→1 x + 1
ex. Evaluate lim
x+3
ex. Evaluate lim 2
.
x→−3 x − 9
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In general, to evaluate limits with
0
indeterminate form we use the following:
0
Suppose f and g are functions such that f (x) = g(x)
when x 6= a.
If lim f (x) = L, then lim g(x) =
x→a
x→a
Rewrite each function and use the above rule to
evaluate the following limits:
x2 − x − 6
ex. lim
x→3 3x − x2
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√
ex. lim
x→3
x − 12 − x
x−3
1
1
−
a
+
h
−
3
a−3
ex. lim
h→0
h
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LIMITS AT INFINITY
x2
Consider the graph of f (x) = 2
:
x +2
What happens to f (x) as x increases in absolute
value?
In other words, what is
lim f (x) =
x→∞
We can also see this from a table of values for
x2
:
f (x) = 2
x +2
x
0 1
10 100
1000
10000
f (x) ≈ 0 .3333 .98 .9998 0.999998 0.99999998
What is lim f (x)?
x→−∞
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Def. Function f has the limit L as x increases
without bound (as x approaches infinity),
written lim f (x) = L, if f (x) can be made as close
x→∞
to L as we want by choosing x large enough.
Similarly we say the function f has the limit M
as x approaches negative infinity,
written lim f (x) = M , if f (x) can be made as
x→−∞
close to M as we want by choosing x negative but
large enough in absolute value.
Def. The line y = L is called a
of the graph of f (x) if either
lim f (x) = L or
x→∞
lim f (x) = L.
x→−∞
x2
has
In our example, the graph of f (x) = 2
x +2
horizontal asymptote
Do all graphs have horizontal asymptotes?
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Consider the following functions:
ex. f (x) = x2
ex. f (x) = 1 − ex
6
6
-
-
?
?
To Evaluate Limits at Infinity
What happens to xn if n is a positive real number
and x → ±∞?
So for a polynomial function
f (x) = anxn + an−1xn−1 + ... + a0, lim f (x) =
x→±∞
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Now consider limits at infinity for
rational functions.
Theorem: Let n be a positive real number and let
k be any nonzero constant. Then
k
=
x→∞ xn
1) lim
k
=
x→−∞ xn
and 2) lim
We use this result and the limit laws (which still apply) to evaluate limits at infinity.
Evaluate each limit, and find any horizontal asymptotes of the following rational functions:
2x + 3
ex. lim 2
x→∞ x − 2x
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2x2 − 2x + 3
ex. Evaluate lim
x→∞
5x2 + 6
2x − x2
ex. Evaluate: 1) lim
x→∞ 2x + 3
2x − x2
2) lim
x→−∞ 2x + 3
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To evaluate limits at infinity and to find
horizontal asymptotes of rational functions:
Given f (x) =
p(x)
.
q(x)
1) If deg p(x) < deg q(x),
2) If deg p(x) = deg q(x),
and
3) If deg p(x) > deg q(x),
3 − x4
ex. Evaluate: lim
x→−∞ (x − 1)(x2 + 2)
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Consider the following limits involving exponentials:
6
-
?
ex.
ex.
lim ex =
x→∞
lim e−x =
x→∞
lim ex =
x→−∞
lim e−x =
x→−∞
ex. Find each horizontal asymptote of the graph of
−3
f (x) =
.
−x
4+e
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Additional Examples
ex.
x+2
x→0+ ln x
lim


 √x − 4 x 6= 4
x−2
ex. Let f (x) =

2
x=4
Find: a) lim f (x)
x→4
b) lim f (x)
x→1
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Now you try it!
1. Evaluate the limits:
1
1
− 2
(a) lim x x − 12
x→4
x−4
√
x+ x+2
(b) lim
x→−1
x+1
x2 + 4x
(c) lim + 2
x→−4 x + 3x − 4
4−x
(d) lim+
x→3 ln(x − 3)
(a + x)2 − a2
(e) lim
x→0
x
2. Let f (x) and g(x) be defined for all real numbers. Which of the
following statements must be true? Which might be true?
Which must be false? Give an example or explain your answer.
(a) lim f (x) = f (a)
x→a
(b) If lim− f (x) = 2 and lim+ f (x) = 1, then lim f (x) = 1.
x→0
x→0
x→0
(c) If lim f (x) = 0 and lim g(x) = 0, then lim [f (x)/g(x)] does not
x→a
x→a
x→a
exist as a finite limit.
3. Evaluate each limit and find any horizontal asymptotes of the corresponding function:
x3 − x
(b) lim 2
x→∞ (x + 2)(2x2 − 1)
s
5 − x2
(d) lim
x→∞
3 − 4x2
3
(a) lim
x→∞ 4 + ln x
x3 + 2x
x→−∞ 2 − x
(c) lim
4. Find each vertical and horizontal asymptote of f (x) =

1
4


−

 x x2
5. For f (x) =
4−x



2
x 6= 4
, find lim f (x).
x→4
x=4
−2
.
4 − ex