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Chapter 4 | Applications of Derivatives
Next we see how to use L’Hôpital’s rule to compare the growth rates of power, exponential, and logarithmic functions.
Example 4.45 Comparing the Growth Rates of ln( x ), x 2 , and e x
For each of the following pairs of functions, use L’Hôpital’s rule to evaluate lim x →∞ ⎛ ⎝ f ( x ) g ( x ) ⎞ ⎠ .
a. f ( x ) = x 2 and g ( x ) = e x b. f ( x ) = ln( x ) and g ( x ) = x 2
Solution
e x =∞, we can use L’Hôpital’s rule to evaluate lim x →∞ ⎡ ⎣ x 2 e x ⎤ ⎦ . We
x 2 =∞ and lim x →∞
a. Since lim x →∞
obtain
x 2 e x = lim x →∞
2 x e x .
lim x →∞
e x =∞, we can apply L’Hôpital’s rule again. Since lim x →∞ 2 x e x = lim x →∞ 2 e x =0,
Since lim x
→∞ 2
x =∞ and lim x →∞
we conclude that
lim x →∞ x 2 e x =0. Therefore, e x grows more rapidly than x 2 as x →∞ (See Figure 4.73 and Table 4.9 ).
Figure 4.73 An exponential function grows at a faster rate than a power function.
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