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Question Number 34375 by rahul 19 last updated on 05/May/18

$$\mathit{T}hevalueof$$$${lim}_{x\to \frac{\pi }{2}}\frac{\left[\frac{x}{2}\right]}{\log \left(\sin x\right)}=?$$$$[.]=greatestintegerfunction.$$

Answered by MJS last updated on 05/May/18

$$f\left(x\right)=\left[\frac{x}{2}\right]=0\mathrm{if}x\mathrm{is}\mathrm{close}\mathrm{to}\frac{\pi }{2}$$$$\mathrm{so}{f}^{\prime }\left(x\right)=0$$$$g\left(x\right)=\ln \sin x$$$$g^{\prime }\left(x\right)=\frac{1}{\tan x}$$$$\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}=0\times \tan x=0$$$$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\left[\frac{x}{2}\right]}{\ln \sin x}=\underset{x\to \frac{\pi }{2}}{\lim 0}\times \tan x=0$$

Commented by rahul 19 last updated on 05/May/18

$$\mathcal{T}hankyousir.$$

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