<s>[INST] Solve limit $\lim_{x\to 0} \frac{x - \sin (x)}{(x \sin (x))^{3/2}}$ Hi I must  solve the next limit $\infty -  $\infty$ usibg L'Hopital and Taylor series. 
$\lim\limits_{x \to 0} \frac{x -  \sin (x)}{(x \sin (x))^{3/2} }$
I tried to eliminate the root with multiplying by 
$1 = \frac{(x \sin(x))^{\frac{1}{2}})}{(x \sin (x))^{\frac{1}{2}}} $
Byt whe I tried to apply L'Hopital the Sin function is very problematic
 [/INST] You have with Taylor series around $0$ : $$\sin(x)=x-\frac{x^3}{6}+o(x^3)$$
$$\lim\limits_{x \to 0} \frac{x -  \sin (x)}{(x \sin (x))^{3/2} }=\lim\limits_{x \to 0} \frac{x -  x + \frac{x^3}{6}+o(x^3)}{(x (x+o(x))^{3/2} }=\lim\limits_{x \to 0} \frac{\frac{x^3}{6}+o(x^3)}{x^3 +o(x^3) }= \frac{1}{6}$$
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