If $\lim _{x \rightarrow 0} \sin \left(\frac{\pi\left(1-\cos ^n x\right)}{x^n}\right)$ exists, where

Question

If $\lim _{x \rightarrow 0} \sin \left(\frac{\pi\left(1-\cos ^n x\right)}{x^n}\right)$ exists, where $m, n \in N$, then the sum of all possible values of $n$ is $\_\_\_\_$

Professor PC Thomas Classes · Answer

$\lim _{x ightarrow 0}\left(\frac{\pi\left(1-\cos ^m xight)}{x^n}ight)$ ∵ use $L \cdot H$. Rule \ $ \begin{aligned} & \Rightarrow \lim _{x ightarrow 0 .}\left(\frac{m \cos ^{m-1} x \cdot \sin x}{n \cdot x^{n-1}}ight) \ & \Rightarrow \frac{m}{n} \lim _{x ightarrow 0} \frac{\sin x}{x^{n-1}} \Rightarrow n-1=0 	ext { or } 1 \ & \Rightarrow n=1 	ext { or } 2 \quad \quad 	ext { for limit to exist } \end{aligned} $