Try putting it in polar coordinates. , of course, and so and so your "R/||(x,y- (00)||" is .
The advantage of polar coordinates here is that the "closeness" to (0, 0) is measured by r alone. No matter what is, that goes to 0 as r goes to 0.
I am trying to show that the following function is differentiable at (0,0):
f (x,y) :
((x^4 + y^4) / (x^2 + y^2)) + 1 if (x,y) =/= (0,0)
1 if (x, y) = (0,0)
I have found the Linear Approximation, L at (0 0) = 1
and that R at (0 0) = f (x, y) - L = (x^4 + y^4) / (x^2 + y^2)
and finally, that R / ||(x,y) - (0,0)|| = ((x^4 + y^4) / (x^2 + y^2)) / sqrt(x^2 + y^2)
Now, I need to show that the limit as (x,y) -> (0,0) of R / ||(x,y) - (0,0)|| = 0
Which brings me to my actual problem...I cannot figure out how to go about solving this limit. If anyone can point me in the right direction I would appreciate it.