Learning Outcomes
- Calculate the limit of a function as π₯ increases or decreases without bound
- Recognize a horizontal asymptote on the graph of a function
We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity. We have looked at vertical asymptotes in other modules; in this section, we deal with horizontal and oblique asymptotes.
Limits at Infinity and Horizontal Asymptotes
Recall that limxβaf(x)=Llimxβaf(x)=L means f(x)f(x) becomes arbitrarily close to LL as long as xx is sufficiently close to aa. We can extend this idea to limits at infinity. For example, consider the function f(x)=2+1xf(x)=2+1x. As can be seen graphically in Figure 1 and numerically in the table beneath it, as the values of xx get larger, the values of f(x)f(x) approach 2. We say the limit as xx approaches ββ of f(x)f(x) is 2 and write limxββf(x)=2limxββf(x)=2. Similarly, for x<0x<0, as the values |x||x| get larger, the values of f(x)f(x) approaches 2. We say the limit as xx approaches ββββ of f(x)f(x) is 2 and write limxβaf(x)=2limxβaf(x)=2.

Figure 1. The function approaches the asymptote y=2y=2 as xx approaches Β±βΒ±β.
xx | 10 | 100 | 1,000 | 10,000 |
2+1x2+1x | 2.1 | 2.01 | 2.001 | 2.0001 |
xx | -10 | -100 | -1000 | -10,000 |
2+1x2+1x | 1.9 | 1.99 | 1.999 | 1.9999 |
More generally, for any function ff, we say the limit as xββxββ of f(x)f(x) is LL if f(x)f(x) becomes arbitrarily close to LL as long as xx is sufficiently large. In that case, we write limxββf(x)=Llimxββf(x)=L. Similarly, we say the limit as xβββxβββ of f(x)f(x) is LL if f(x)f(x) becomes arbitrarily close to LL as long as x<0x<0 and |x||x| is sufficiently large. In that case, we write limxβββf(x)=Llimxβββf(x)=L. We now look at the definition of a function having a limit at infinity.
Definition
(Informal) If the values of f(x)f(x) become arbitrarily close to LL as xx becomes sufficiently large, we say the function ff has a limit at infinity and write
If the values of f(x)f(x) becomes arbitrarily close to LL for x<0x<0 as |x||x| becomes sufficiently large, we say that the function ff has a limit at negative infinity and write
If the values f(x)f(x) are getting arbitrarily close to some finite value LL as xββxββ or xβββxβββ, the graph of ff approaches the line y=Ly=L. In that case, the line y=Ly=L is a horizontal asymptote of ff (Figure 2). For example, for the function f(x)=1xf(x)=1x, since limxββf(x)=0limxββf(x)=0, the line y=0y=0 is a horizontal asymptote of f(x)=1xf(x)=1x.
Definition
If limxββf(x)=Llimxββf(x)=L or limxβββf(x)=Llimxβββf(x)=L, we say the line y=Ly=L is a horizontal asymptote of ff.

Figure 2. (a) As xββxββ, the values of ff are getting arbitrarily close to LL. The line y=Ly=L is a horizontal asymptote of ff. (b) As xβββxβββ, the values of ff are getting arbitrarily close to MM. The line y=My=M is a horizontal asymptote of ff.
A function cannot cross a vertical asymptote because the graph must approach infinity (or negative infinity) from at least one direction as xx approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function f(x)=cosxx+1f(x)=cosxx+1 shown in Figure 3 intersects the horizontal asymptote y=1y=1 an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.

Figure 3. The graph of f(x)=cosx/x+1f(x)=cosx/x+1 crosses its horizontal asymptote y=1y=1 an infinite number of times.
The algebraic limit laws and squeeze theorem we introduced in Why It Matters: Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.
Example: Computing Limits at Infinity
For each of the following functions ff, evaluate limxββf(x)limxββf(x) and limxβββf(x)limxβββf(x). Determine the horizontal asymptote(s) for ff.
- f(x)=5β2x2f(x)=5β2x2
- f(x)=sinxxf(x)=sinxx
- f(x)=tanβ1(x)f(x)=tanβ1(x)
Watch the following video to see the worked solution to Example: Computing Limits at Infinity.
Try It
Evaluate limxβββ(3+4x)limxβββ(3+4x) and limxββ(3+4x)limxββ(3+4x). Determine the horizontal asymptotes of f(x)=3+4xf(x)=3+4x, if any.
Try It
Infinite Limits at Infinity
Sometimes the values of a function ff become arbitrarily large as xββxββ (or as xβββ)xβββ). In this case, we write limxββf(x)=βlimxββf(x)=β (or limxβββf(x)=β)limxβββf(x)=β). On the other hand, if the values of ff are negative but become arbitrarily large in magnitude as xββxββ (or as xβββ)xβββ), we write limxββf(x)=ββlimxββf(x)=ββ (or limxβββf(x)=ββ)limxβββf(x)=ββ).
For example, consider the function f(x)=x3f(x)=x3. As seen in the table below and Figure 8, as xββxββ the values f(x)f(x) become arbitrarily large. Therefore, limxββx3=βlimxββx3=β. On the other hand, as xβββxβββ, the values of f(x)=x3f(x)=x3 are negative but become arbitrarily large in magnitude. Consequently, limxβββx3=ββlimxβββx3=ββ.
xx | 10 | 20 | 50 | 100 | 1000 |
x3x3 | 1000 | 8000 | 125,000 | 1,000,000 | 1,000,000,000 |
xx | -10 | -20 | -50 | -100 | -1000 |
x3x3 | -1000 | -8000 | -125,000 | -1,000,000 | -1,000,000,000 |

Figure 8. For this function, the functional values approach infinity as xβΒ±βxβΒ±β.
Definition
(Informal) We say a function ff has an infinite limit at infinity and write
if f(x)f(x) becomes arbitrarily large for xx sufficiently large. We say a function has a negative infinite limit at infinity and write
if f(x)<0f(x)<0 and |f(x)||f(x)| becomes arbitrarily large for xx sufficiently large. Similarly, we can define infinite limits as xβββxβββ.
Formal Definitions
Earlier, we used the terms arbitrarily close, arbitrarily large, and sufficiently large to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.
Definition
(Formal) We say a function ff has a limit at infinity, if there exists a real number LL such that for all Ξ΅>0Ξ΅>0, there exists N>0N>0 such that
for all x>Nx>N. In that case, we write
(see Figure 9).
We say a function ff has a limit at negative infinity if there exists a real number LL such that for all Ξ΅>0Ξ΅>0, there exists N<0N<0 such that
for all [latex]x

Figure 9. For a function with a limit at infinity, for all x>Nx>N, |f(x)βL|<Ξ΅|f(x)βL|<Ξ΅.
Earlier in this section, we used graphical evidence and numerical evidence to conclude that limxββ(2+1x)=2limxββ(2+1x)=2. Here we use the formal definition of limit at infinity to prove this result rigorously.
Example: A Finite Limit at Infinity Example
Use the formal definition of limit at infinity to prove that limxββ(2+1x)=2.
Watch the following video to see the worked solution to Example: A Finite Limit at Infinity Example.
Try It
Use the formal definition of limit at infinity to prove that limxββ(3β1x2)=3.
We now turn our attention to a more precise definition for an infinite limit at infinity.
Definition
(Formal) We say a function f has an infinite limit at infinity and write
if for all M>0, there exists an N>0 such that
for all x>N (see Figure 10).
We say a function has a negative infinite limit at infinity and write
if for all M<0, there exists an N>0 such that
for all x>N.
Similarly we can define limits as xβββ.

Figure 10. For a function with an infinite limit at infinity, for all x>N, f(x)>M.
Earlier, we used graphical evidence (Figure 8) and numerical evidence (the table beneath it) to conclude that limxββx3=β. Here we use the formal definition of infinite limit at infinity to prove that result.
Example: An Infinite Limit at Infinity
Use the formal definition of infinite limit at infinity to prove that limxββx3=β.
Watch the following video to see the worked solution to Example: An Infinite Limit at Infinity.
Try It
Use the formal definition of infinite limit at infinity to prove that limxββ3x2=β.
Try It
Candela Citations
- 4.6 Limits at Infinity and Asymptotes. Authored by: Ryan Melton. License: CC BY: Attribution
- Calculus Volume 1. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/details/books/calculus-volume-1. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction