Growth Rates of Functions

Learning Outcomes

  • Describe the relative growth rates of functions

Suppose the functions ff and gg both approach infinity as xx. Although the values of both functions become arbitrarily large as the values of xx become sufficiently large, sometimes one function is growing more quickly than the other. For example, f(x)=x2f(x)=x2 and g(x)=x3g(x)=x3 both approach infinity as xx. However, as shown in the following table, the values of x3x3 are growing much faster than the values of x2x2.

Comparing the Growth Rates of x2x2 and x3x3
xx 10 100 1000 10,000
f(x)=x2f(x)=x2 100 10,000 1,000,000 100,000,000
g(x)=x3g(x)=x3 1000 1,000,000 1,000,000,000 1,000,000,000,0001,000,000,000,000

In fact,

limxx3x2=limxx=limxx3x2=limxx=  or, equivalently, limxx2x3=limx1x=0limxx2x3=limx1x=0

 

As a result, we say x3x3 is growing more rapidly than x2x2 as xx. On the other hand, for f(x)=x2f(x)=x2 and g(x)=3x2+4x+1g(x)=3x2+4x+1, although the values of g(x)g(x) are always greater than the values of f(x)f(x) for x>0x>0, each value of g(x)g(x) is roughly three times the corresponding value of f(x)f(x) as xx, as shown in the following table. In fact,

limxx23x2+4x+1=13limxx23x2+4x+1=13

 

Comparing the Growth Rates of x2x2 and 3x2+4x+13x2+4x+1
xx 10 100 1000 10,000
f(x)=x2f(x)=x2 100 10,000 1,000,000 100,000,000
g(x)=3x2+4x+1g(x)=3x2+4x+1 341 30,401 3,004,001 300,040,001

In this case, we say that x2x2 and 3x2+4x+13x2+4x+1 are growing at the same rate as xx.

More generally, suppose ff and gg are two functions that approach infinity as xx. We say gg grows more rapidly than ff as xx if

limxg(x)f(x)=limxg(x)f(x)=  or, equivalently, limxf(x)g(x)=0limxf(x)g(x)=0

 

On the other hand, if there exists a constant M0M0 such that

limxf(x)g(x)=Mlimxf(x)g(x)=M,

 

we say ff and gg grow at the same rate as xx.

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

example: Comparing the Growth Rates of lnxlnx, x2x2, and exex

For each of the following pairs of functions, use L’Hôpital’s rule to evaluate limx(f(x)g(x))limx(f(x)g(x)).

  1. f(x)=x2f(x)=x2 and g(x)=exg(x)=ex
  2. f(x)=lnxf(x)=lnx and g(x)=x2g(x)=x2

Watch the following video to see the worked solution to Example: Comparing the Growth Rates of lnxlnx, x2x2, and exex.

Try It

Compare the growth rates of x100x100 and 2x2x.

Using the same ideas as in the last example. it is not difficult to show that exex grows more rapidly than xpxp for any p>0p>0. In Figure 5 and the table below it, we compare exex with x3x3 and x4x4 as xx.

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 exex grows faster than xpxp for any p>0p>0. (a) A comparison of exex with x3x3. (b) A comparison of exex with x4x4.

An exponential function grows at a faster rate than any power function
xx 5 10 15 20
x3x3 125 1000 3375 8000
x4x4 625 10,000 50,625 160,000
exex 148 22,026 3,269,017 485,165,195

Similarly, it is not difficult to show that xpxp grows more rapidly than lnxlnx for any p>0p>0. In Figure 6 and the table below it, we compare lnxlnx with 3x3x and xx.

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 y=lnxy=lnx grows more slowly than xpxp for any p>0p>0 as xx.

A logarithmic function grows at a slower rate than any root function
xx 10 100 1000 10,000
lnxlnx 2.303 4.605 6.908 9.210
3x3x 2.154 4.642 10 21.544
xx 3.162 10 31.623 100