The value of x 0lim2(1- x 2x√3 3x x^2) is __________.

Question

The value of $\underset{x	o 0}{lim}2(\frac{1-\cos x\sqrt{\cos 2x}\sqrt3\cos 3x}{\ldots }{x}^{2})$ is \_\_\_\_\_\_\_\_\_\_.

Answer: 55

Accepted Answer

$\underset{x	o 0}{lim}2(\frac{1-\cos x{(\cos 2x)}^{\frac{1}{2}}{(\cos 3x)}^{\frac{1}{3}}\, ...\, {(\cos 10x)}^{\frac{1}{10}}}{{x}^{2}})$ $(\frac{0}{0})$

Using L' hospital

$2\underset{x	o 0}{lim}\frac{\sin x(\cos 2x{)}^{\frac{1}{2}}\ldots (\cos 10x{)}^{\frac{1}{10}}\ldots (\sin 2x)(\cos x)(\cos 3x{)}^{\frac{1}{3}}+\ldots }{2x}$

$\begin{gathered} \Rightarrow \underset{x	o 0}{lim}(\frac{\sin x}{x}+\frac{\sin 2x}{x}+\ldots +\frac{\sin 10x}{x}) \ \, =1+2+\ldots +10=55 \end{gathered}$

Solutions · 1