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 · 2017-12-11T09:03:15

Please refer to a Proof in the Explanation.

#"Reqd. Lim. L="lim_(theta to pi/4)(sintheta-costheta)/(theta-pi/4).#

Let, #x=theta-pi/4. :." As "theta to pi/4 rArr x to 0.#

#:. L=lim_(x to 0){sin(x+pi/4)-cos(x+pi/4)}/x,#

#=lim1/x{sinxcos(pi/4)+cosxsin(pi/4)-cosxcos(pi/4)+sinxsin(pi/4)},#

#=lim1/x{1/sqrt2(sinx+cosx-cosx+sinx)},#

#=lim_(x to 0)(2sinx)/(sqrt2x),#

Since, #lim_(alpha to 0)sinalpha/alpha=1,# we have,

#L=2/sqrt2*1=sqrt2.#

Q.E.D.