<s>[INST] Trigonometrical limit $$\lim\limits_{x \to 0} \frac{x^4\cos(\frac{1}{x})} {\sin^3x},$$
I have a problem with this limit. I tried to use L'Hôpital's rule but it is not effective. Please help.
 [/INST] $$\lim\limits_{x → 0} \frac{x^4 \cos \left(x^{-1}\right)}{\sin ^3\left(x\right)} \times \frac{x^{-3}}{x^{-3}}$$

$$\lim\limits_{x → 0} \frac{x\cos \left(x^-1\right)}{x^{-3}\sin ^3\left(x\right)}$$

$$\lim\limits_{x → 0} \frac{ \frac{\cos \left(x^{-1}\right)}{x^{-1}} }{ \frac{\sin ^3\left(x\right)}{x^3} } $$


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*Using $$\lim\limits_{x → 0} \frac{\sin \left(x\right)} {x}=1 \quad \textrm{ & } \lim\limits_{x → 0} \frac{\cos \left(1/x\right)} {1/x}=0$$

$$\therefore \frac{ \lim\limits_{x → 0} 
\frac{\cos \left(x^{-1}\right)}{x^{-1}} }{ \lim\limits_{x → 0} \left( \frac{\sin \left(x\right)}{x} \right)^3 } =0$$
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