Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 121527 by benjo_mathlover last updated on 09/Nov/20

lim_(x→0) ((sin x cos x)/( (√(π+2sin x)) −(√π))) ?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\pi+\mathrm{2sin}\:\mathrm{x}}\:−\sqrt{\pi}}\:? \\ $$

Answered by Dwaipayan Shikari last updated on 09/Nov/20

lim_(x→0) ((xcosx)/(π+2sinx−π))((√(π+2sinx))+(√π)) =(x/(2sinx))((√π)+(√π)) (sinx→x and x→0) =((2(√π))/2)=(√π)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{xcosx}}{\pi+\mathrm{2}{sinx}−\pi}\left(\sqrt{\pi+\mathrm{2}{sinx}}+\sqrt{\pi}\right) \\ $$$$=\frac{{x}}{\mathrm{2}{sinx}}\left(\sqrt{\pi}+\sqrt{\pi}\right)\:\:\:\:\:\:\:\:\:\left({sinx}\rightarrow{x}\:\:\:{and}\:{x}\rightarrow\mathrm{0}\right) \\ $$$$=\frac{\mathrm{2}\sqrt{\pi}}{\mathrm{2}}=\sqrt{\pi} \\ $$

Answered by liberty last updated on 09/Nov/20

lim_(x→0) ((sin x cos x)/( (√π) ((√(1+((2sin x)/π)))−1 ))) = lim_(x→0) ((cos x)/( (√π))). ((sin x)/((1+((sin x)/π))−1)) = (1/( (√π))) × lim_(x→0) ((π sin x)/(sin x)) = (π/( (√π))) = (√π) .

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\pi}\:\left(\sqrt{\mathrm{1}+\frac{\mathrm{2sin}\:\mathrm{x}}{\pi}}−\mathrm{1}\:\right)}\:= \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\pi}}.\:\frac{\mathrm{sin}\:\mathrm{x}}{\left(\mathrm{1}+\frac{\mathrm{sin}\:\mathrm{x}}{\pi}\right)−\mathrm{1}}\:=\: \\ $$$$\frac{\mathrm{1}}{\:\sqrt{\pi}}\:×\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\pi\:\mathrm{sin}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}}\:=\:\frac{\pi}{\:\sqrt{\pi}}\:=\:\sqrt{\pi}\:.\: \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com