### Home > APCALC > Chapter 6 > Lesson 6.1.4 > Problem 6-53

6-53.

Evaluate. Homework Help ✎

Be careful! Evaluating this limit as

*n*→ 0 leads to trouble:That means we do not know if the limit is finite (because the 0s '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:

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.