Let $f(x, y) = \dfrac{x^n + y^n}{x^2 - y^2}$, $n > 2$, be defined in $U = \left\{(x, y) \in \mathbb{R}^2 : y \neq x, y \neq -x\right\}$. My problem is to determine the existence of $\lim\limits_{\substack{(x, y) \rightarrow (0, 0) \\ (x, y) \in U}} f(x, y)$.
The solution offered by a professor is the following:
Let $V = \left\{(x, x + x^p) : x > 0\right\}$, $p > n - 1$. If $(x, y) \in V$ then$$f(x, y) = f\left(x, x + x^p\right) = \dfrac{x^n + \left(x + x^p\right)^n}{x^2 - \left(x + x^p\right)^2} = -\dfrac{x^n + x^n + nx^{n - 1 + p} + \cdots + x^{np}}{2x^{p + 1} + x^{2p}}.$$As $p > n - 1$, if $x \rightarrow 0$ then $f\left(x, x + x^p\right) \rightarrow \infty$. Thus, $f$ is not convergent when $(x, y) \rightarrow (0, 0)$ in $U$.
My question is how could I come up with this idea. Firstly, I'd have to think about the non-existence instead of the existence, and still I wouldn't know how to get to the set $V$ in order to prove it. Moreover, is there another way to do it?