Section 2.6: Limits Involving Infinity and Asymptotes.
We start this section by looking at the function f (x) = 1/x. Using the graphical approach, compute the
following limits:
1
=
x→−∞ x
1
=
x→∞ x
lim
lim
As a side note, if your function is 1/xn , with n > 0, Then
1
=
x→−∞ xn
lim
1
=
x→∞ xn
and
lim
We then follow this up with a definition:
A function f (x) has a horizontal asymptote of y = b (with b a real number) if either:
lim f (x) = b
or
x→−∞
lim f (x) = b
x→∞
Examples: Use the notes above to determine the horizontal asymptotes of the following functions:
√
8x3 + 2x − 1
x−5
6x3 + 3x2 − 2x − 9
√
g(x)
=
h(x)
=
f (x) =
3
5x3 − 7
x2 − 4
x−5
Conceptually, can you come up with a non rational function that has a horizontal asymptote on the x-axis?
Another type of asymptote is what’s called a slant asymptote. By definition, a function f (x) has a slant
asymptote at y = mx + b if either:
lim [f (x) − (mx + b)] = 0
lim [f (x) − (mx + b)] = 0
or
x→−∞
x→∞
Example: Compute the slant asymptote of the function below:
f (x) =
We again consider the function f (x) =
2x3 − 2x + 9
x2 + x − 2
1
.
x
Notice that as x approaches 0 from the left, the graph is going to −∞, while as x approaches 0 from the
right, the graph is going to +∞. This gives us the characteristic criteria for a vertical asymptote. More
precisely, the line x = a is a vertical asymptote of a function f (x) if either:
lim f (x) = +∞ or − ∞
x→a+
or
lim f (x) = +∞ or − ∞
x→a−
As a note here: limits that evaluate to infinity technically do not exist, since ±∞ is not a real number.
When looking for vertical asymptotes, you’re looking for x = a values that makes the limit go to either
+∞ or −∞. Polynomials will never have vertical asymptotes because lim f (x) = f (a) for all real numbers.
x→a
Rational functions can have vertical asymptotes at x = a, where a makes the denominator equal to zero.
But again, you have to be careful that it meets the criteria. See the example below:
Example: Consider the function f (x) =
x2
x
. Compute all of its vertical asymptotes.
− 2x
Some conceptual notes asymptotes:
1. For a rational function r(x) =
n(x)
:
d(x)
(a) If the degree of n(x) = the degree of d(x), then the horizontal asymptote is y = p/q, with p
and q being the leading coefficients of n(x) and d(x), respectively.
(b) If the degree of n(x) > the degree of d(x), then you will not have a horizontal asymptote.
(c) If the degree of n(x) < the degree of d(x), then you will have a horizontal asymptote at y = 0.
(d) If the degree of n(x) is exactly one higher than the degree of d(x), then you will have a slant
asymptote.
(e) The line x = a is a vertical asymptote of r(x) if d(a) = 0, but n(a) 6= 0.
NOTE: These rules do not necessarily apply if you are not working with a rational function!
2. If a function f (x) has a slant asymptote, it cannot have a horizontal asymptote.
3. All linear functions have a slant asymptote, namely itself.
4. Functions can have infinitely many vertical asymptotes. (Can you name one that does?)
5. Asymptotes are lines. It does not make sense to say a function f (x) has an asymptote at 3.
Exercises: Compute all asymptotes of the following rational functions:
1. f (x) =
5x2 + 9x + 1
8x2 + 1
2. g(x) =
x+2
x4 − 16
3. h(x) =
6x2 + 10x + 10
x−2
4. j(x) =
x4 + 2x + 1
x3 − 1
5. k(x) =
x4 + 1
x2 − 3x