L’Hôpital’s Rule: Learn It 4

Growth Rates of Functions

Suppose the functions [latex]f[/latex] and [latex]g[/latex] both approach infinity as [latex]x\to \infty[/latex]. Although the values of both functions become arbitrarily large as the values of [latex]x[/latex] become sufficiently large, sometimes one function is growing more quickly than the other.

For example, [latex]f(x)=x^2[/latex] and [latex]g(x)=x^3[/latex] both approach infinity as [latex]x\to \infty[/latex]. However, as shown in the following table, the values of [latex]x^3[/latex] are growing much faster than the values of [latex]x^2[/latex].

Comparing the Growth Rates of [latex]x^2[/latex] and [latex]x^3[/latex]
[latex]x[/latex] [latex]10[/latex] [latex]100[/latex] [latex]1000[/latex] [latex]10,000[/latex]
[latex]f(x)=x^2[/latex] [latex]100[/latex] [latex]10,000[/latex] [latex]1,000,000[/latex] [latex]100,000,000[/latex]
[latex]g(x)=x^3[/latex] [latex]1000[/latex] [latex]1,000,000[/latex] [latex]1,000,000,000[/latex] [latex]1,000,000,000,000[/latex]

In fact,

[latex]\underset{x\to \infty }{\lim}\dfrac{x^3}{x^2}=\underset{x\to \infty}{\lim} x=\infty[/latex]  or, equivalently, [latex]\underset{x\to \infty}{\lim}\dfrac{x^2}{x^3}=\underset{x\to \infty }{\lim}\dfrac{1}{x}=0[/latex]

As a result, we say [latex]x^3[/latex] is growing more rapidly than [latex]x^2[/latex] as [latex]x\to \infty[/latex].

On the other hand, for [latex]f(x)=x^2[/latex] and [latex]g(x)=3x^2+4x+1[/latex], although the values of [latex]g(x)[/latex] are always greater than the values of [latex]f(x)[/latex] for [latex]x>0[/latex], each value of [latex]g(x)[/latex] is roughly three times the corresponding value of [latex]f(x)[/latex] as [latex]x\to \infty[/latex], as shown in the following table. In fact,

[latex]\underset{x\to \infty }{\lim}\dfrac{x^2}{3x^2+4x+1}=\dfrac{1}{3}[/latex]
Comparing the Growth Rates of [latex]x^2[/latex] and [latex]3x^2+4x+1[/latex]
[latex]x[/latex] [latex]10[/latex] [latex]100[/latex] [latex]1000[/latex] [latex]10,000[/latex]
[latex]f(x)=x^2[/latex] [latex]100[/latex] [latex]10,000[/latex] [latex]1,000,000[/latex] [latex]100,000,000[/latex]
[latex]g(x)=3x^2+4x+1[/latex] [latex]341[/latex] [latex]30,401[/latex] [latex]3,004,001[/latex] [latex]300,040,001[/latex]

In this case, we say that [latex]x^2[/latex] and [latex]3x^2+4x+1[/latex] are growing at the same rate as [latex]x\to \infty[/latex].

More generally, suppose [latex]f[/latex] and [latex]g[/latex] are two functions that approach infinity as [latex]x\to \infty[/latex]. We say [latex]g[/latex] grows more rapidly than [latex]f[/latex] as [latex]x\to \infty[/latex] if

[latex]\underset{x\to \infty }{\lim}\dfrac{g(x)}{f(x)}=\infty[/latex]  or, equivalently, [latex]\underset{x\to \infty }{\lim}\dfrac{f(x)}{g(x)}=0[/latex]

On the other hand, if there exists a constant [latex]M \ne 0[/latex] such that

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

we say [latex]f[/latex] and [latex]g[/latex] grow at the same rate as [latex]x\to \infty[/latex].

Next we see how to use L’Hôpital’s rule to compare the growth rates of power, exponential, and logarithmic functions.

For each of the following pairs of functions, use L’Hôpital’s rule to evaluate [latex]\underset{x\to \infty }{\lim}\left(\dfrac{f(x)}{g(x)}\right)[/latex].

  1. [latex]f(x)=x^2[/latex] and [latex]g(x)=e^x[/latex]
  2. [latex]f(x)=\ln x[/latex] and [latex]g(x)=x^2[/latex]

Using the same ideas as in the last example. it is not difficult to show that [latex]e^x[/latex] grows more rapidly than [latex]x^p[/latex] for any [latex]p>0[/latex]. In Figure 5 and the table below it, we compare [latex]e^x[/latex] with [latex]x^3[/latex] and [latex]x^4[/latex] as [latex]x\to \infty[/latex].

This figure has two figures marked a and b. In figure a, the functions y = ex and y = x3 are graphed. It is obvious that ex increases more quickly than x3. In figure b, the functions y = ex and y = x4 are graphed. It is obvious that ex increases much more quickly than x4, but the point at which that happens is further to the right than it was for x3.
Figure 5. The exponential function [latex]e^x[/latex] grows faster than [latex]x^p[/latex] for any [latex]p>0[/latex]. (a) A comparison of [latex]e^x[/latex] with [latex]x^3[/latex]. (b) A comparison of [latex]e^x[/latex] with [latex]x^4[/latex].
An exponential function grows at a faster rate than any power function
[latex]x[/latex] [latex]5[/latex] [latex]10[/latex] [latex]15[/latex] [latex]20[/latex]
[latex]x^3[/latex] [latex]125[/latex] [latex]1000[/latex] [latex]3375[/latex] [latex]8000[/latex]
[latex]x^4[/latex] [latex]625[/latex] [latex]10,000[/latex] [latex]50,625[/latex] [latex]160,000[/latex]
[latex]e^x[/latex] [latex]148[/latex] [latex]22,026[/latex] [latex]3,269,017[/latex] [latex]485,165,195[/latex]

Similarly, it is not difficult to show that [latex]x^p[/latex] grows more rapidly than [latex]\ln x[/latex] for any [latex]p>0[/latex]. In Figure 6 and the table below it, we compare [latex]\ln x[/latex] with [latex]\sqrt[3]{x}[/latex] and [latex]\sqrt{x}[/latex].

This figure shows y = the square root of x, y = the cube root of x, and y = ln(x). It is apparent that y = ln(x) grows more slowly than either of these functions.
Figure 6. The function [latex]y=\ln x[/latex] grows more slowly than [latex]x^p[/latex] for any [latex]p>0[/latex] as [latex]x\to \infty[/latex].
A logarithmic function grows at a slower rate than any root function
[latex]x[/latex] [latex]10[/latex] [latex]100[/latex] [latex]1000[/latex] [latex]10,000[/latex]
[latex]\ln x[/latex] [latex]2.303[/latex] [latex]4.605[/latex] [latex]6.908[/latex] [latex]9.210[/latex]
[latex]\sqrt[3]{x}[/latex] [latex]2.154[/latex] [latex]4.642[/latex] [latex]10[/latex] [latex]21.544[/latex]
[latex]\sqrt{x}[/latex] [latex]3.162[/latex] [latex]10[/latex] [latex]31.623[/latex] [latex]100[/latex]