Let F: $\displaystyle R^2 \rightarrow R$ be defined by:

Determine if the $\displaystyle \lim_{(x,y) \to (0,0)}$ exists, if it does prove it.

Here's my attempt:

As (x,y) -> (0,0) along the y-axis, x=0:

$\displaystyle \lim_{(x,y) \to (0,0)} y^2 sin(\frac{1}{y^2})$

Along the x-axis:

$\displaystyle \lim_{(x,y) \to (0,0)} x^2 sin(\frac{1}{x^2})$

Along the line y=x:

$\displaystyle \lim_{(x,y) \to (0,0)} (x^2 +x) sin(\frac{1}{x^2 +x^2})$

I tried showing whether there is a common limit along different paths but I don't know how to finish this. Any help here is appreciated.