Limit of dirichlet function (from DSP)

[Jyan]

How is this limit evaluated?

$$\lim_{k->0}\frac{sin(\pi k)}{sin(\frac{\pi k}{N})}$$

I know that it is N, but I can't figure out how to evaluate it, l'hospital's rule doesn't seem to help.

I might solve it by the time I get a response, but figured no reason to not ask especially since I couldn't find much about it on Google.

Solved it, feel like an idiot:

$$\lim_{k->0}\frac{sin(\pi k)}{sin(\frac{\pi k}{N})}$$

Using l'hospital's rule:

$$\lim_{k->0}N\frac{cos(\pi k)}{cos(\frac{\pi k}{N})}$$

This is equal to N, since cos(0) = 1.

 

[JJacquelin]

Series expansion of the sin function is an easy way to evaluate the limit.
In using the Landau symbol : http://mathworld.wolfram.com/AsymptoticNotation.html