<s>[INST] value of $n$ in limits 
If $\displaystyle \lim_{x\rightarrow 0}\frac{x^n\sin^{n}(x)}{x^{n}-(\sin x)^{n}}$ is a finite non zero number.
Then value of $n$ is equals

What I try:
$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots $$
$$\lim_{x\rightarrow 0}\frac{x^n\bigg(x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \bigg)^n}{x^{n}-\bigg(x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \bigg)^n}$$
$$\lim_{x\rightarrow 0}\frac{x^{n+1}\bigg(1-\frac{x^2}{3!}+\frac{x^4}{5!}+\cdots \bigg)^n}{x^{n}-\bigg(x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \bigg)^n}$$
How do I solve it? Help me please.
 [/INST] $$\frac{\sin x}{x}\simeq 1-\frac{1}{6}x^2+\frac{1}{120}x^4+O(x^6)$$
$$\left(\frac{\sin x}{x}\right)^n\simeq 1-\frac{n}{6}x^2+\frac{n(5n-2)}{360}x^4+O(x^6)$$
$$\frac{\left(\frac{\sin x}{x}\right)^n}{1-\left(\frac{\sin x}{x}\right)^n}\simeq \frac{6}{n}x^{-2}-\frac{5n+2}{10n}+O(x^2)$$
$$\boxed{f(x)=\frac{x^n\sin^n x}{x^n-sin^nx}\simeq\frac{6}{n}x^{n-2}-\frac{5n+2}{10n}x^n+O(x^{n+2})}$$
If $\quad n<2\qquad$ $f(x)\sim\frac{6}{n}\frac{1}{x^{2-n}}\to\infty$
If $\quad n=2\qquad f(x)\sim 3-\frac35 x^2\to 3$ 
If $\quad n>2\qquad f(x)\sim\frac{6}{n}x^{n-2}\to 0 $
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