LIM=(TGX-SINX)/(3X^2) ПРИ X=0

Решите задачу:

$L=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{3x^2}\\\\L=\dfrac{1}{3}\lim\limits_{x\to 0}\dfrac{(1-\cos x)\sin x}{x^2\cos x}\\\\L=\dfrac{1}{3}\lim\limits_{x\to 0}\dfrac{1-\cos x}{x}\cdot\dfrac{\sin x}{x}\cdot\dfrac{1}{\cos x}\\\\L=\dfrac{1}{3}\lim\limits_{x\to 0}\dfrac{(1-\cos x)(1+\cos x)}{x(1+\cos x)}\cdot1\cdot1\\\\L=\dfrac{1}{3}\lim\limits_{x\to 0}\dfrac{1-\cos^2 x}{x(1+\cos x)}\\\\$

$L = \dfrac{1}{3} \lim \limits_{x \to 0} \dfrac{\sin ^ 2 x}{x (1+ \cos x)} \\\\L = \dfrac{1}{3} \lim \limits_{x \to 0} \dfrac{\sin x}{1+ \cos x} \cdot \dfrac{\sin x}{x} \\\\L = \dfrac{1}{3} \lim \limits_{x \to 0} \dfrac{\sin x}{1+ \cos x} \cdot 1 \\\\L = \dfrac{1}{3} \lim \limits_{x \to 0} \dfrac{\sin x}{1+ \cos x} \\\\L = \dfrac{1}{3} \cdot \dfrac{0}{2} \\\\\boxed{L = 0}$