Differentiability of a two variable function with parameter

  • #1

Homework Statement



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

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:

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

(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):

[tex]\lim_{(h,k)\to (0, 0)}\frac{f(h, k)}{\sqrt{h^2+k^2}}[/tex] 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 :)

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
SammyS
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See if the limit of each of the following exists as (x, y) → (0, 0):
[itex]\displaystyle \frac{\partial f(x,\,y)}{\partial x}[/itex] and [itex]\displaystyle \frac{\partial f(x,\,y)}{\partial y}[/itex]​
 
Last edited:

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