Question Number 34375 by rahul 19 last updated on 05/May/18
![The value of lim_(x→(π/2)) (([(x/2)])/(log (sin x))) = ? [.]= greatest integer function.](Q34375.png)
$$\boldsymbol{{T}}{he}\:{value}\:{of}\: \\ $$$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \:\:\:\frac{\left[\frac{{x}}{\mathrm{2}}\right]}{\mathrm{log}\:\left(\mathrm{sin}\:{x}\right)}\:=\:? \\ $$$$\left[.\right]=\:{greatest}\:{integer}\:{function}. \\ $$
Answered by MJS last updated on 05/May/18
![f(x)=[(x/2)]=0 if x is close to (π/2) so f′(x)=0 g(x)=ln sin x g′(x)=(1/(tan x)) ((f′(x))/(g′(x)))=0×tan x=0 lim_(x→(π/2)) (([(x/2)])/(ln sin x))=lim_(x→(π/2)) 0×tan x=0](Q34378.png)
$${f}\left({x}\right)=\left[\frac{{x}}{\mathrm{2}}\right]=\mathrm{0}\:\mathrm{if}\:{x}\:\mathrm{is}\:\mathrm{close}\:\mathrm{to}\:\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{so}\:{f}'\left({x}\right)=\mathrm{0} \\ $$$${g}\left({x}\right)=\mathrm{ln}\:\mathrm{sin}\:{x} \\ $$$${g}'\left({x}\right)=\frac{\mathrm{1}}{\mathrm{tan}\:{x}} \\ $$$$\frac{{f}'\left({x}\right)}{{g}'\left({x}\right)}=\mathrm{0}×\mathrm{tan}\:{x}=\mathrm{0} \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\frac{\left[\frac{{x}}{\mathrm{2}}\right]}{\mathrm{ln}\:\mathrm{sin}\:{x}}=\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}0}×\mathrm{tan}\:{x}=\mathrm{0} \\ $$
Commented by rahul 19 last updated on 05/May/18
$$\mathscr{T}{hank}\:{you}\:{sir}. \\ $$