<s>[INST] Evaluating $\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$ What is the value of
$$\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$$
I have already tried the two paths result and I think now that the limit does exist and it is equals to 0. But i really cannot prove this.
 [/INST] 
$$L=\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2-y^2}$$

$$L=\lim_{(x, y) \to (0, 0)} \frac{xy^2-x^3+x^3}{x^2-y^2}=\lim_{x \to0} (-x)+\lim_{(x, y) \to (0, 0)} \frac{x^3}{x^2-y^2}=\lim_{(x, y) \to (0, 0)} \frac{x^3}{x^2-y^2}$$
Let $y=x-ax^2$
$$L=\lim_{x \to 0} \frac{x^3}{2ax^3-a^2x^4}=\lim_{x \to 0} \frac{1}{2a-a^2x}=\frac{1}{2a}$$
So the limit depends on the value of $a$, hence the limit doesn't exist.
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