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 haveΒ 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 [latex]\underset{x \to a}{\lim}f(x)=L[/latex] means [latex]f(x)[/latex] becomes arbitrarily close to [latex]L[/latex] as long as [latex]x[/latex] is sufficiently close to [latex]a[/latex]. We can extend this idea to limits at infinity. For example, consider the function [latex]f(x)=2+\frac{1}{x}[/latex]. As can be seen graphically in Figure 1 and numerically in the table beneath it, as the values of [latex]x[/latex] get larger, the values of [latex]f(x)[/latex] approach 2. We say the limit as [latex]x[/latex] approaches [latex]\infty [/latex] of [latex]f(x)[/latex] is 2 and write [latex]\underset{x\to \infty }{\lim}f(x)=2[/latex]. Similarly, for [latex]x<0[/latex], as the values [latex]|x|[/latex] get larger, the values of [latex]f(x)[/latex] approaches 2. We say the limit as [latex]x[/latex] approaches [latex]β\infty [/latex] of [latex]f(x)[/latex] is 2 and write [latex]\underset{x\to a}{\lim}f(x)=2[/latex].
[latex]x[/latex] | 10 | 100 | 1,000 | 10,000 |
[latex]2+\frac{1}{x}[/latex] | 2.1 | 2.01 | 2.001 | 2.0001 |
[latex]x[/latex] | -10 | -100 | -1000 | -10,000 |
[latex]2+\frac{1}{x}[/latex] | 1.9 | 1.99 | 1.999 | 1.9999 |
More generally, for any function [latex]f[/latex], we say the limit as [latex]x \to \infty [/latex] of [latex]f(x)[/latex] is [latex]L[/latex] if [latex]f(x)[/latex] becomes arbitrarily close to [latex]L[/latex] as long as [latex]x[/latex] is sufficiently large. In that case, we write [latex]\underset{x\to \infty}{\lim}f(x)=L[/latex]. Similarly, we say the limit as [latex]x\to β\infty [/latex] of [latex]f(x)[/latex] is [latex]L[/latex] if [latex]f(x)[/latex] becomes arbitrarily close to [latex]L[/latex] as long as [latex]x<0[/latex] and [latex]|x|[/latex] is sufficiently large. In that case, we write [latex]\underset{x\to β\infty }{\lim}f(x)=L[/latex]. We now look at the definition of a function having a limit at infinity.
Definition
(Informal) If the values of [latex]f(x)[/latex] become arbitrarily close to [latex]L[/latex] as [latex]x[/latex] becomes sufficiently large, we say the function [latex]f[/latex] has a limit at infinity and write
If the values of [latex]f(x)[/latex] becomes arbitrarily close to [latex]L[/latex] for [latex]x<0[/latex] as [latex]|x|[/latex] becomes sufficiently large, we say that the function [latex]f[/latex] has a limit at negative infinity and write
If the values [latex]f(x)[/latex] are getting arbitrarily close to some finite value [latex]L[/latex] as [latex]x\to \infty [/latex] or [latex]x\to β\infty[/latex], the graph of [latex]f[/latex] approaches the line [latex]y=L[/latex]. In that case, the line [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex] (Figure 2). For example, for the function [latex]f(x)=\frac{1}{x}[/latex], since [latex]\underset{x\to \infty }{\lim}f(x)=0[/latex], the line [latex]y=0[/latex] is a horizontal asymptote of [latex]f(x)=\frac{1}{x}[/latex].
Definition
If [latex]\underset{x\to \infty }{\lim}f(x)=L[/latex] or [latex]\underset{x \to β\infty}{\lim}f(x)=L[/latex], we say the line [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex].
A function cannot cross a vertical asymptote because the graph must approach infinity (or negative infinity) from at least one direction as [latex]x[/latex] 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 [latex]f(x)=\frac{ \cos x}{x}+1[/latex] shown in Figure 3 intersects the horizontal asymptote [latex]y=1[/latex] an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.
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 [latex]f[/latex], evaluate [latex]\underset{x\to \infty }{\lim}f(x)[/latex] and [latex]\underset{x\to β\infty }{\lim}f(x)[/latex]. Determine the horizontal asymptote(s) for [latex]f[/latex].
- [latex]f(x)=5-\frac{2}{x^2}[/latex]
- [latex]f(x)=\dfrac{\sin x}{x}[/latex]
- [latex]f(x)= \tan^{-1} (x)[/latex]
Watch the following video to see the worked solution to Example: Computing Limits at Infinity.
Try It
Evaluate [latex]\underset{x\to β\infty}{\lim}\left(3+\frac{4}{x}\right)[/latex] and [latex]\underset{x\to \infty }{\lim}\left(3+\frac{4}{x}\right)[/latex]. Determine the horizontal asymptotes of [latex]f(x)=3+\frac{4}{x}[/latex], if any.
Try It
Infinite Limits at Infinity
Sometimes the values of a function [latex]f[/latex] become arbitrarily large as [latex]x\to \infty [/latex] (or as [latex]x\to β\infty )[/latex]. In this case, we write [latex]\underset{x\to \infty }{\lim}f(x)=\infty [/latex] (or [latex]\underset{x\to β\infty }{\lim}f(x)=\infty )[/latex]. On the other hand, if the values of [latex]f[/latex] are negative but become arbitrarily large in magnitude as [latex]x\to \infty [/latex] (or as [latex]x\to β\infty )[/latex], we write [latex]\underset{x\to \infty }{\lim}f(x)=β\infty [/latex] (or [latex]\underset{x\to β\infty }{\lim}f(x)=β\infty )[/latex].
For example, consider the function [latex]f(x)=x^3[/latex]. As seen in the table below and Figure 8, as [latex]x\to \infty [/latex] the values [latex]f(x)[/latex] become arbitrarily large. Therefore, [latex]\underset{x\to \infty }{\lim}x^3=\infty[/latex]. On the other hand, as [latex]x\to β\infty[/latex], the values of [latex]f(x)=x^3[/latex] are negative but become arbitrarily large in magnitude. Consequently, [latex]\underset{x\to β\infty }{\lim}x^3=β\infty[/latex].
[latex]x[/latex] | 10 | 20 | 50 | 100 | 1000 |
[latex]x^3[/latex] | 1000 | 8000 | 125,000 | 1,000,000 | 1,000,000,000 |
[latex]x[/latex] | -10 | -20 | -50 | -100 | -1000 |
[latex]x^3[/latex] | -1000 | -8000 | -125,000 | -1,000,000 | -1,000,000,000 |
Definition
(Informal) We say a function [latex]f[/latex] has an infinite limit at infinity and write
if [latex]f(x)[/latex] becomes arbitrarily large for [latex]x[/latex] sufficiently large. We say a function has a negative infinite limit at infinity and write
if [latex]f(x)<0[/latex] and [latex]|f(x)|[/latex] becomes arbitrarily large for [latex]x[/latex] sufficiently large. Similarly, we can define infinite limits as [latex]x\to β\infty[/latex].
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 [latex]f[/latex] has a limit at infinity, if there exists a real number [latex]L[/latex] such that for all [latex]\varepsilon >0[/latex], there exists [latex]N>0[/latex] such that
for all [latex]x>N[/latex]. In that case, we write
(see Figure 9).
We say a function [latex]f[/latex] has a limit at negative infinity if there exists a real number [latex]L[/latex] such that for all [latex]\varepsilon >0[/latex], there exists [latex]N<0[/latex] such that
for all [latex]x<N[/latex]. In that case, we write
Earlier in this section, we used graphical evidence and numerical evidence to conclude that [latex]\underset{x\to \infty }{\lim}\left(2+\frac{1}{x}\right)=2[/latex]. 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 [latex]\underset{x\to \infty }{\lim}\left(2+\frac{1}{x}\right)=2[/latex].
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 [latex]\underset{x\to \infty }{\lim}\left(3 - \dfrac{1}{x^2}\right)=3[/latex].
We now turn our attention to a more precise definition for an infinite limit at infinity.
Definition
(Formal) We say a function [latex]f[/latex] has an infinite limit at infinity and write
if for all [latex]M>0[/latex], there exists an [latex]N>0[/latex] such that
for all [latex]x>N[/latex] (see Figure 10).
We say a function has a negative infinite limit at infinity and write
if for all [latex]M<0[/latex], there exists an [latex]N>0[/latex] such that
for all [latex]x>N[/latex].
Similarly we can define limits as [latex]x\to β\infty[/latex].
Earlier, we used graphical evidence (Figure 8) and numerical evidence (the table beneath it) to conclude that [latex]\underset{x\to \infty }{\lim}x^3=\infty[/latex]. 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 [latex]\underset{x\to \infty }{\lim}x^3=\infty[/latex].
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 [latex]\underset{x\to \infty }{\lim}3x^2=\infty[/latex].