Finding that a two variable limit does not exist trouble: $\lim_{(x, y) \to (0, 0)} \frac{y^2 \sin^2 x}{x^4 + y^4}$ I have
$$ \lim_{(x, y) \to (0, 0)} \frac{y^2 \sin^2 x}{x^4 + y^4} $$
I've tried setting $y= 0, x=0, y=mx, y=x$, and lots of other things but because of the $\sin$ I keep getting a limit of zero. How do you show this doesn't exist?

As you say: if you let $x = 0$ or $y = 0$ your answer will be 0. If you let $y = x$, however, you get
\begin{equation*}
\lim_{x \rightarrow 0} \frac{x^2 \sin^2(x)}{x^4 + x^4}
= \lim_{x \rightarrow 0} \frac{\sin^2(x)}{2x^2}
= \lim_{x \rightarrow 0} \frac{\sin(x)}{x} \cdot \frac{\sin(x)}{x} \cdot \frac{1}{2} = \frac{1}{2}.
\end{equation*}
