Thread: Show function is differentiable over its domain

I need to show that:



is differentiable over its domain. The only point I see there being a problem at is when (x,y,z) = (0,0,0).

Then is it sufficient to show that:



where for this case.

I do see a problem in the h(x,y,z) function regarding f(0,0,0). That's what I am trying to find the limit of, and in my function I'd get 0/0 when I plug in those values. Also, I am getting the partial derivatives to be of the form:





Using all the info I seem to get many terms who wind up becoming 0/0. I believe I am missing the point and/or missing a way to simplify everything. Thanks for any help.

I just realized that the functions domain is:

all R3 except for (0,0,0).

The function isn't defined at that point and so it isn't in its domain.

Does that mean I can simply use the above formula to show that for all (a,b,c) the limit exists? If that's the case then I suppose I would need to show that the limit = 0 for all (a, b, c) correct? The reason is that for anyway I look at the limit at (0,0,0) I appear to be getting that it doesn't exist.