Question Number 60410 by tanmay last updated on 20/May/19
![lim_(x→0) [(x^2 /(tanxsinx))] [.]=grestest integer function](Q60410.png)
$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{{x}^{\mathrm{2}} }{{tanxsinx}}\right]\:\left[.\right]={grestest}\:{integer}\:{function} \\ $$
Commented by kaivan.ahmadi last updated on 20/May/19
![if 0<x<(π/2) sinx<x<tgx⇒xsinx<x^2 <xtgx⇒ ((xsinx)/(sinxtgx))<(x^2 /(sinxtgx))<((xtgx)/(sinxtgx))⇒ (x/(tgx))<(x^2 /(sinxtgx))<(x/(sinx))⇒ [(x/(tgx))]<[(x^2 /(sinxtgx))]<[(x/(sinx))]⇒ 0<[(x^2 /(sinxtgx))]<1⇒ lim_(x→0) [(x^2 /(sinxtgx))]=0](Q60417.png)
$${if}\:\mathrm{0}<{x}<\frac{\pi}{\mathrm{2}} \\ $$$${sinx}<{x}<{tgx}\Rightarrow{xsinx}<{x}^{\mathrm{2}} <{xtgx}\Rightarrow \\ $$$$\frac{{xsinx}}{{sinxtgx}}<\frac{{x}^{\mathrm{2}} }{{sinxtgx}}<\frac{{xtgx}}{{sinxtgx}}\Rightarrow \\ $$$$\frac{{x}}{{tgx}}<\frac{{x}^{\mathrm{2}} }{{sinxtgx}}<\frac{{x}}{{sinx}}\Rightarrow \\ $$$$\left[\frac{{x}}{{tgx}}\right]<\left[\frac{{x}^{\mathrm{2}} }{{sinxtgx}}\right]<\left[\frac{{x}}{{sinx}}\right]\Rightarrow \\ $$$$\mathrm{0}<\left[\frac{{x}^{\mathrm{2}} }{{sinxtgx}}\right]<\mathrm{1}\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} \:\:\left[\frac{{x}^{\mathrm{2}} }{{sinxtgx}}\right]=\mathrm{0} \\ $$$$ \\ $$
Commented by tanmay last updated on 20/May/19
$${excllent} \\ $$