1. Now we are concerned about limits and we can have nightmarish limits. For example if
What can we do?
We end up using derivatives to help evaluate such limits. We have L’Hopital’s rule state, if , then
2. Sketch of Proof. We can use linear approximations writing
Use these approximations in (1a), then take the limit . So our limit becomes
We evaluate the inner limit first
But look , so (4) simplifies to
Thus we obtain
So we obtain
3. Sketch of Proof for . When
we run into similar problems. L’Hopital’s rule still holds. We just do the following trick:
and we recover the previous case…kind of!
Taking the derivative of top and bottom, we get
We can cross multiply, giving us
Cross multiplying gives us
Notation: We will write
when we apply L’Hopital’s rule.
Example 1. Consider the sequence
What happens as ? We have to apply L’Hopital’s rule twice
Thus we have