Question

Evaluate the limit by recognizing each as the definition of a derivative

lim (as z approaches e) (lnz-1)/(z-e)

Answer 1

This limit is by definition, the derivative of ln z evaluated at z = e.

So instead of evaluating the limit explicitly, write down the derivative of ln z and evaluate it at z = e.

Answer 2

L'Hôpital's rule works

Answer 3

Using l"hopital's rule here is logically a bit perverse, since logically l'hopital's rule requires you to know the derivative of ln z, which result is exactly the limit you are trying to find.

Answer 4

Thanks!

Answer 5

I feel the same way when people say that one should evaluate
\[\lim_{x \rightarrow 0} {\sin x \over x} = 1\]
use l'Hopital's rule. Well if you know how to show the derivative of sin is cos, this limit is required, so it would be circular reasoning.

(The best intuitive proof of this limit for me is geometric. Draw a unit circle and look at what happens to sin x and x, as measured by arc length.)