|
|
L=\mathop {\lim }\limits_{x \to 0} \dfrac{1 - \cos^{3} x}{x\sin x} =\mathop {\lim }\limits_{x \to 0} \dfrac{1 - \cos x}{x\sin x} (1+\cos x+\cos^{2} x)
=\mathop {\lim }\limits_{x \to 0} \dfrac{1 - \cos x}{x\sin x} .\mathop {\lim }\limits_{x \to 0}(1+\cos x+\cos^{2} x)
=\mathop {\lim }\limits_{x \to 0} \dfrac{2\sin^2\frac{x}{2}}{x\sin x}. (1+1+1 )
= \dfrac{3}{2}\mathop {\lim }\limits_{x \to 0} \dfrac{\sin^2\frac{x}{2}}{\left ( \frac{x}{2} \right )^2}. \mathop {\lim }\limits_{x \to 0} \dfrac{x}{\sin x}
= \dfrac{3}{2}.1.1
= \dfrac{3}{2}
|