SOLUTION:.
BEWARE: L’Hôpital’s Rule applies only to indeterminate forms and . Trying
to use it in other situations leads to incorrect results, like this:
For more practice, redo the Practice Exercises at the end of Chapter 2,
applying L’Hôpital’s Rule wherever possible.
NOTE: Below is a description of how to determine the limit for other
indeterminate forms by transforming them into or , but only limits originally
of the form and will be tested on the AP Calculus exam as given in Examples
38–44. Examples 45, 46, and 47 are presented to complete the discussion of
indeterminate forms and L’Hôpital’s Rule, but questions similar to those in
Examples 45–47 will not appear on the AP Calculus exam.
L’Hôpital’s Rule can be applied also to indeterminate forms of the types 0 · ∞
and ∞ − ∞, if the forms can be transformed to either or .
Example 45 __
Find .
SOLUTION: is of the type ∞ · 0. Since and, as x → ∞,
the latter is the indeterminate form , we see that
(Note the easier solution
Other indeterminate forms, such as 0^0 , 1∞, and ∞^0 , may be resolved by taking
the natural logarithm and then applying L’Hôpital’s Rule.
Example 46 __
Find .