Differentiability of a two variable function with parameter

[Mathitalian]

Homework Statement

For which parameter \alpha\in\mathbb{R} the function:
f(x, y)= \begin{cases}|x|^\alpha \sin(y),&\mbox{ if } x\ne 0;\\ 0, & \mbox{ if } x=0\end{cases}

is differentiable at the point (0, 0)?

The Attempt at a Solution

For α<0, the function is not continue at (0, 0), so it is not differentiable. I checked it :)

For α≥0 i have troubles, many troubles. I need your help.

I started evalueting the partial derivates using the definition:

\partial_x f(0,0):= \lim_{h\to 0} \frac{|h|^\alpha\sin(0)-0}{h}=0
\partial_y f(0,0):= \lim_{h\to 0} \frac{|0|^\alpha\sin(h)-0}{h}=0

(I think i must split the cases α=0 and α>0, right? |0|^α has no sense if α=0, but if α=0 then f(x,y)= sin(y)... I'm confuse)

Anyway, i need to find the value of the limit (if it exists):

\lim_{(h,k)\to (0, 0)}\frac{f(h, k)}{\sqrt{h^2+k^2}} and show that its value is 0 right? I have problems with two variables limits...

Any helps will be appreciated...

Please, if you see there are mistakes in English languange, correct me :)

 

[SammyS]

​

See if the limit of each of the following exists as (x, y) → (0, 0):

\displaystyle \frac{\partial f(x,\,y)}{\partial x} and \displaystyle \frac{\partial f(x,\,y)}{\partial y}​