Finding the limit of a function with a complex exponent

[Grothard]

Homework Statement

\lim_{z \to 0} (\frac{sinz}{z})^{1/z^2}

where z is complex

Homework Equations

The standard definition of a limit
L'hopital's rule?

The Attempt at a Solution

I'm quite stumped by this one. There doesn't seem to be a way to break it down into different limits or even to manipulate it much algebraically. Wolfram says the answer is e^{-1/6} but I'm not sure how to arrive at it. I tried starting by proving that that is the limit if z is real and then extending it to the complex plane, but I can't even solve it for that case. I feel like there's an obvious theorem or something along those lines that I am forgetting. Any hints?

 

[gb7nash]

1) log it
2) find the limit. It helps to know what (sinz)/z approaches as z approaches 0. Can we apply l'hopital's rule?
3) reverse the log.

 

[Grothard]

It worked! Thanks, that was a really clever and elegant solution