How do I sketch this kind of function? Do I find what the limit is first? $$F(x)=\lim_{n\to\infty}\frac{x^{2n}\sin\left(\dfrac{nx}{2}\right)+x^2}{x^{2n}+1}$$
I know that the extreme values are $0$ and $2$, and the function diverges by oscillating boundedly.
Any help would be appreciated.
We have that for $|x|<1\: F(x)\to x^2$ and for $|x|>1$
$$\frac{x^{2n}\sin\left(\frac{nx}{2}\right)+x^2}{x^{2n}+1}= \frac{x^{2n}}{x^{2n}}\frac{\sin\left(\frac{nx}{2}\right)+x^{2-2n}}{1+x^{-2n}}\sim \sin\left(\frac{nx}{2}\right) $$
therefore the limit doesn’t exist since for any $x\neq 2k\pi$ the $\sin$ term oscillates between $-1$ and $1$ as also for the special case $|x|=1$.