Answer: First, rewrite this expression as . Notice that when x nears infinity, the expression becomes
, which is indeterminate.Take the derivative of the top and bottom.
= 0
PROBLEM 3. Find .
Answer: We learned this in Chapter 3, remember? Now we’ll use L’Hôpital’s Rule. At first glance, the
limit is indeterminate: . Let’s take some derivatives.
This is still indeterminate, so it’s time to take the derivative of the top and bottom again.
It’s still indeterminate! If you try it one more time, you’ll get a fraction with no variables: , which can
be simplified to (as we expected).
PROBLEM 4. Find .
Answer: Plugging in gives you the indeterminate response of . The derivative is
When we take the limit of this expression, we get −1.