TITLE: Testing continuity of the function $f(x) = \lim\limits_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \ \text{for} \ x \in \mathbb{R}$
QUESTION [10 upvotes]: My Question is:

Examine the continuity of $$f(x) = \lim_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \qquad \text{for} \ x \in \mathbb{R}$$

How can I do this? Honestly, speaking I have $\text{no idea}$ of doing, this as this seems to be a tough limit. What I only know is that as $n \to \infty$ $\sin{n}$ $\text{oscillates}$ between $-1$ and $1$.

REPLY [6 votes]: Proceeding from where Arturo left note that:

$|2\sin{x}|< 1$ $\Longrightarrow$ $x \in \displaystyle\Bigl(n\pi-\frac{\pi}{6},n\pi+\frac{\pi}{6}\:\Bigr)$
$|2\sin{x}|=1 \Longrightarrow \displaystyle x =n\pi+\frac{\pi}{6}$
$|2\sin{x}|>1 \Longrightarrow \displaystyle x \in \Bigl(n\pi + \frac{\pi}{6}, n\pi+\frac{5\pi}{6}\:\Bigr)$

So your $f(x)$ is basically, $$ f(x) = \left\{\begin{array}{lll} x & \text{if}\ x\in  \Bigl(n\pi-\frac{\pi}{6},n\pi+\frac{\pi}{6}\Bigr) \\\ \frac{x}{2} & \text{if} \ x =n\pi+\frac{\pi}{6} \\\ 0 & \text{if} \ x \in \Bigl(n\pi + \frac{\pi}{6}, n\pi+\frac{5\pi}{6}\Bigr)\end{array}\right.$$
It is easy to note $\text{that the only points which gives us trouble are}$ $x =n\pi \pm \frac{\pi}{6}$. Now it is your job is to check the continuity of $f(x)$ at $x=n\pi \pm \frac{\pi}{6}$.