By definition it is discontinuous if the following is not true:
$$\lim_{x \to a} f(x) = f(a)$$
I can only go as far as simplifying the function a bit, which is not enough. I thought that it would be discontinuous if: 1) the denominator is zero. That never happens in this case. 2) the numerator and denominator both go to zero. Also never happens. 3) the numerator and denominator are both infinities, that may happen.
Simplification:
$$f(x) = \lim_{n \to \infty} \frac{(x^{n}-1)(x^{n}+1)}{x^{2n}+2}$$ So does not help me much.
You can explicity determine what $f(x)$ is for $x \in\mathbb{R}$. For example if $|x|<1$, then $x^n\to 0$ whence $$ \frac{x^{2n}-1}{x^{2n}+2}\to \frac{-1}{2}. $$ If $|x|>1$, then $1/x^n\to 0$ whence $$ \frac{x^{2n}-1}{x^{2n}+2}=\frac{1-x^{-2n}}{1+2x^{-2n}}\to 1. $$ If $x=1, -1$, the limit is zero. Hence $$ f(x)=\begin{cases} -1/2 &\quad |x|<1\\ 1&\quad |x|>1\\ 0&\quad |x|=1. \end{cases} $$ You should be able to take it from here.