Now, the limit is . The derivative of the top and bottom is
Don’t you wish that you had learned this back when you first did limits?
Example 6: Find .
The derivative of the top and bottom is
Example 7: Find x cot x.
Taking this limit results in (0)(∞), which is also indeterminate. (But you can’t use the rule yet!) If you
rewrite this expression as , it’s of the form , and we can use L’Hôpital’s Rule.
That’s all that you need to know about L’Hôpital’s Rule. Just check to see if the limit results in an
indeterminate form. If it does, use the rule until you get a determinate form.
Here are some more examples.
PROBLEM 1. Find .
Answer: First, notice that plugging in 0 gives us an indeterminate result: . Now, take the derivative of
the top and of the bottom.
PROBLEM 2. Find xe−2x.