Limits at Infinity

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]βˆ’\infty [/latex] of [latex]f(x)[/latex] is 2 and write [latex]\underset{x\to a}{\lim}f(x)=2[/latex].

The function f(x) 2 + 1/x is graphed. The function starts negative near y = 2 but then decreases to βˆ’βˆž near x = 0. The function then decreases from ∞ near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.

Figure 1. The function approaches the asymptote [latex]y=2[/latex] as [latex]x[/latex] approaches [latex]\pm \infty[/latex].

Values of a function [latex]f[/latex] as [latex]x \to \pm \infty [/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 βˆ’\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 βˆ’\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

[latex]\underset{x\to \infty }{\lim}f(x)=L[/latex]

 

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

[latex]\underset{x\to -\infty }{\lim}f(x)=L[/latex]

 

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 βˆ’\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 βˆ’\infty}{\lim}f(x)=L[/latex], we say the line [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex].

The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.

Figure 2. (a) As [latex]x\to \infty[/latex], the values of [latex]f[/latex] are getting arbitrarily close to [latex]L[/latex]. The line [latex]y=L[/latex] is a horizontal asymptote of [latex]f[/latex]. (b) As [latex]x\to βˆ’\infty[/latex], the values of [latex]f[/latex] are getting arbitrarily close to [latex]M[/latex]. The line [latex]y=M[/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 function f(x) = (cos x)/x + 1 is shown. It decreases from (0, ∞) and then proceeds to oscillate around y = 1 with decreasing amplitude.

Figure 3. The graph of [latex]f(x)=\cos x/x+1[/latex] crosses its horizontal asymptote [latex]y=1[/latex] 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 [latex]f[/latex], evaluate [latex]\underset{x\to \infty }{\lim}f(x)[/latex] and [latex]\underset{x\to βˆ’\infty }{\lim}f(x)[/latex]. Determine the horizontal asymptote(s) for [latex]f[/latex].

  1. [latex]f(x)=5-\frac{2}{x^2}[/latex]
  2. [latex]f(x)=\dfrac{\sin x}{x}[/latex]
  3. [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 βˆ’\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 βˆ’\infty )[/latex]. In this case, we write [latex]\underset{x\to \infty }{\lim}f(x)=\infty [/latex] (or [latex]\underset{x\to βˆ’\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 βˆ’\infty )[/latex], we write [latex]\underset{x\to \infty }{\lim}f(x)=βˆ’\infty [/latex] (or [latex]\underset{x\to βˆ’\infty }{\lim}f(x)=βˆ’\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 βˆ’\infty[/latex], the values of [latex]f(x)=x^3[/latex] are negative but become arbitrarily large in magnitude. Consequently, [latex]\underset{x\to βˆ’\infty }{\lim}x^3=βˆ’\infty[/latex].

Values of a power function as [latex]x\to \pm \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
The function f(x) = x3 is graphed. It is apparent that this function rapidly approaches infinity as x approaches infinity.

Figure 8. For this function, the functional values approach infinity as [latex]x\to \pm \infty[/latex].

Definition


(Informal) We say a function [latex]f[/latex] has an infinite limit at infinity and write

[latex]\underset{x\to \infty }{\lim}f(x)=\infty [/latex]

 

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

[latex]\underset{x\to \infty }{\lim}f(x)=βˆ’\infty[/latex]

 

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 βˆ’\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

[latex]|f(x)-L|<\varepsilon [/latex]

 

for all [latex]x>N[/latex]. In that case, we write

[latex]\underset{x\to \infty }{\lim}f(x)=L[/latex]

 

(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

[latex]|f(x)-L|<\varepsilon [/latex]

 

for all [latex]x<N[/latex]. In that case, we write

[latex]\underset{x\to βˆ’\infty }{\lim}f(x)=L[/latex]

 

The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + ΰ₯‰ and L – ΰ₯‰. On the x axis, N is marked as the value of x such that f(x) = L + ΰ₯‰.

Figure 9. For a function with a limit at infinity, for all [latex]x>N[/latex], [latex]|f(x)-L|<\varepsilon [/latex].

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

[latex]\underset{x\to \infty }{\lim}f(x)=\infty [/latex]

 

if for all [latex]M>0[/latex], there exists an [latex]N>0[/latex] such that

[latex]f(x)>M[/latex]

 

for all [latex]x>N[/latex] (see Figure 10).

 

We say a function has a negative infinite limit at infinity and write

[latex]\underset{x\to \infty }{\lim}f(x)=βˆ’\infty [/latex]

 

if for all [latex]M<0[/latex], there exists an [latex]N>0[/latex] such that

[latex]f(x)<M[/latex]

 

for all [latex]x>N[/latex].

Similarly we can define limits as [latex]x\to βˆ’\infty[/latex].

The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.

Figure 10. For a function with an infinite limit at infinity, for all [latex]x>N[/latex], [latex]f(x)>M[/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].

Try It