Question

Evaluate, `lim_(xto(pi//4)) (sinx-cosx)/(x-pi/4)`

Answer 1

Given, `underset(xto(pi/4))"lim"(sqrt(2)(sinx.1/(sqrt(2))-cosx.1/(sqrt(2))))/((x-x/4))=underset(xto(pi//4))"lim"(sqrt(2)sinxcospi/4-cosx.sinpi/4)/(x-pi/4)`
`=underset(xto(pi//4))"lim"(sqrt(2){sin(x-pi/4))}/(x-pi/4)`
`=sqrt(2)underset(xto(pi//4))"lim" (sin(x-pi/4))/(x-pi/4)=sqrt(2)` `[therefore underset(xto0)"lim"(sinx)/x=1]`