Putting x=theta-pi/4 show that limit of (sintheta-costheta)/(theta-pi/4)=sqrt2 as theta approaches to pi/4?

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Answer 1 · 2018-04-02T18:10:53

#L=sqrt2#

Here,

#L=lim_(theta to pi/4)(sintheta-costheta)/(theta-pi/4)#

Now,

#sintheta-costheta=sintheta-sin(pi/2- theta)=2cos((theta+pi/2-theta)/2)sin((theta-pi/2+theta)/2)#

#=2cos(pi/4)sin(theta-pi/4)#

#=2*1/sqrt2sin(theta-pi/4)#

#=sqrt2sin(theta-pi/4)#

So,

#L=lim_(theta to pi/4)(sqrt2sin(theta-pi/4))/(theta-pi/4)#

#=>L=sqrt2*lim_((theta -pi/4)to0)sin(theta-pi/4)/(theta- pi/4)=sqrt2*1#

#=>L=sqrt2#