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Question Number 60410 by tanmay last updated on 20/May/19

$$\underset{x\to 0}{\lim }\left[\frac{x^2}{tanxsinx}\right][.]=grestestintegerfunction$$

Commented by kaivan.ahmadi last updated on 20/May/19

$$if0<x<\frac{\pi }{2}$$$$sinx<x<tgx\Rightarrow xsinx<x^2<xtgx\Rightarrow$$$$\frac{xsinx}{sinxtgx}<\frac{x^2}{sinxtgx}<\frac{xtgx}{sinxtgx}\Rightarrow$$$$\frac{x}{tgx}<\frac{x^2}{sinxtgx}<\frac{x}{sinx}\Rightarrow$$$$\left[\frac{x}{tgx}\right]<\left[\frac{x^2}{sinxtgx}\right]<\left[\frac{x}{sinx}\right]\Rightarrow$$$$0<\left[\frac{x^2}{sinxtgx}\right]<1\Rightarrow$$$${lim}_{x\to 0}\left[\frac{x^2}{sinxtgx}\right]=0$$$$$$

Commented by tanmay last updated on 20/May/19

$$excllent$$

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