Be careful! Evaluating this limit as
leads to trouble:
That means we do not know if the limit is finite (because the
s 'cancel out') or infinite (because there is a in the denominator).
We can evaluate an indeterminate limit if that limit happens to be a definition of the derivative.
Sure enough, part (a) is Ana's Definition of the Derivative:
, notice that . So we can rewrite the limit as:
So, we just need to find
at and we have evaluated the limit.
Now evaluate that derivative at
Refer to part (a).
l’Hôpital’s Rule can be used.
See part (c).
Be sure to reevaluate the limit after each use of l’Hôpital’s Rule.