I have this limit for $x\to n\pi$ with $n\in\mathbb{N}$; the parameter $g$ is real. Numerically, I see that, in general, the limit for $x\to n\pi$ is indeterminate since the left and right limits are not equal but both tend to infinity (plus or minus).
My question. Is it possible to show that regardless of the direction of the limit (left and right limits), this limit will always tend to infinity, either plus or minus infinity?
$$ \lim_{x\to (n\pi)} \frac{1}{\sin x}\sqrt{\left( g\sin x+\cos x\right)^2-1} $$