We have , so if is differentiable at the only possible differential is . Now, analyze if .
I came to that lim term myself and that's where I kind of get stuck with all the epsilons. I can't draw a concrete conclusion from there if it is indeed 0 or just a small epsilon bigger than 0. That's what making this exercise difficult.
I came to a conclusion that the function is indeed differentiable, but I have to show a strict rigorous proof.
How do I expand f to its Taylor Series?
And from there, how do I find the tangent plane?
That is a correct conclusion.
There are many ways to show a rigorous proof.but I have to show a strict rigorous proof.
You only need for finding in a comfortable way the limit in my answer #2.How do I expand f to its Taylor Series?
Use the well known formula where belongs to the surface.And from there, how do I find the tangent plane?
Sorry for the follow up questions. Even with the Taylor expansion, how do I use it in the lim term you prescribed earlier?
And also, after i prove differentiability, how do I find the appropriate P0 to plug into the formula for the tangent plane?
I'm sorry for asking pathetic questions