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6-53.

Evaluate. Homework Help ✎

  1. 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:

  1. Refer to part (a).

  1. l’Hôpital’s Rule can be used.

  1. See part (c).
    Be sure to reevaluate the limit after each use of l’Hôpital’s Rule.