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Question Number 42781 by maxmathsup by imad last updated on 02/Sep/18

calculate lim_(x→(π/4))       ((sin(2x)sin(x−(π/4)))/(sinx −cosx))

$${calculate}\:{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:\:\:\:\:\:\frac{{sin}\left(\mathrm{2}{x}\right){sin}\left({x}−\frac{\pi}{\mathrm{4}}\right)}{{sinx}\:−{cosx}} \\ $$

Commented by maxmathsup by imad last updated on 04/Oct/18

let A(x)=((sin(2x)sin(x−(π/4)))/(sinx−cosx))  we have A(x)=((sin(2x)sin(x−(π/4)))/((√2)sin(x−(π/4))))  ⇒lim_(x→(π/4))    A(x)=lim_(x→(π/4))    ((sin(2x))/(√2)) =(1/(√2)) .

$${let}\:{A}\left({x}\right)=\frac{{sin}\left(\mathrm{2}{x}\right){sin}\left({x}−\frac{\pi}{\mathrm{4}}\right)}{{sinx}−{cosx}}\:\:{we}\:{have}\:{A}\left({x}\right)=\frac{{sin}\left(\mathrm{2}{x}\right){sin}\left({x}−\frac{\pi}{\mathrm{4}}\right)}{\sqrt{\mathrm{2}}{sin}\left({x}−\frac{\pi}{\mathrm{4}}\right)} \\ $$$$\Rightarrow{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:\:\:{A}\left({x}\right)={lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\sqrt{\mathrm{2}}}\:=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:. \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Sep/18

lim_(x→(Π/4))    ((sin2x×(1/(√2))(sinx−cosx))/(sinx−cosx))  =(1/(√2))×sin(Π/2)=(1/(√2))            lim_(t→0)   li_(t→0)

$${li}\underset{{x}\rightarrow\frac{\Pi}{\mathrm{4}}} {{m}}\:\:\:\frac{{sin}\mathrm{2}{x}×\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\left({sinx}−{cosx}\right)}{{sinx}−{cosx}} \\ $$$$=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}×{sin}\frac{\Pi}{\mathrm{2}}=\frac{\mathrm{1}}{\sqrt{\mathrm{2}}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}} \\ $$$$\underset{{t}\rightarrow\mathrm{0}} {\mathrm{li}} \\ $$

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