Chapter 4 | Applications of Derivatives
459
ln x 5 x
4.38
Evaluate lim x →∞
.
As mentioned, L’Hôpital’s rule is an extremely useful tool for evaluating limits. It is important to remember, however, that to apply L’Hôpital’s rule to a quotient f ( x ) g ( x ) , it is essential that the limit of f ( x ) g ( x ) be of the form 0 0 or ∞/∞. Consider
the following example.
Example 4.40 When L’Hôpital’s Rule Does Not Apply
x 2 +5 3 x +4 .
Consider lim x →1
Show that the limit cannot be evaluated by applying L’Hôpital’s rule.
Solution Because the limits of the numerator and denominator are not both zero and are not both infinite, we cannot apply L’Hôpital’s rule. If we try to do so, we get d dx ⎛ ⎝ x 2 +5 ⎞ ⎠ =2 x and d dx (3 x +4) =3. At which point we would conclude erroneously that lim x →1 x 2 +5 3 x +4 = lim x →1 2 x 3 = 2 3 . However, since lim x →1 ⎛ ⎝ x 2 +5 ⎞ ⎠ =6 and lim x →1 (3 x +4) =7, we actually have lim x →1 x 2 +5 3 x +4 = 6 7 . We can conclude that
d dx ⎛ ⎝ x 2 +5 d dx (3 x +4) . ⎞ ⎠
x 2 +5 3 x +4 ≠ lim x →1
lim x →1
Explain why we cannot apply L’Hôpital’s rule to evaluate lim x →0 + cos x
cos x
4.39
x . Evaluate lim x →0 +
x byother
means.
Other Indeterminate Forms L’Hôpital’s rule is very useful for evaluating limits involving the indeterminate forms 0 0 and ∞/∞. However, we can also use L’Hôpital’s rule to help evaluate limits involving other indeterminate forms that arise when evaluating limits. The expressions 0·∞, ∞−∞, 1 ∞ , ∞ 0 , and 0 0 are all considered indeterminate forms. These expressions are not real numbers. Rather, they represent forms that arise when trying to evaluate certain limits. Next we realize why these are indeterminate forms and then understand how to use L’Hôpital’s rule in these cases. The key idea is that we must rewrite
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