Evaluating the Complex Limit: Proving Existence and Value

[ehrenfest]

[SOLVED] complex limit

Homework Statement

Evaluate the complex limit if it exists:

\lim_{z \to 1} \frac{\log{z}}{z-1}

where log denotes the principal branch of the logarithm.

Homework Equations

The Attempt at a Solution

I am pretty sure it exists and equals 1, because that is what it equals when I approach with specific sequences. But how can I prove that?

 

[Gib Z]

\lim_{z \to 1} \frac{\log{z}}{z-1} = \lim_{z\to 0} \frac{\log (1+z)}{z}.

Are you allowed to use the property that the series for that log term converges to that log term iff |z| < 1?