Be careful! Evaluating this limit as n → 0 leads us to trouble:
Refer to part (a).
That means we do not know if the limit is finite (because the 0's 'cancel out') or infinite (because there is a 0 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:
Since n → 0, notice that a = 0. So we can rewrite the limit as:
So, we just need to find f '(x) at x = 0 and we have evaluated the limit.
Now evaluate that derivative at x = 0: