Question

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

Answer 1

Please refer to a Proof in the Explanation.

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.