The value of `lim_(xto(1)/(sqrt(2))) (x-cos(sin^(-1)x))/(1-tan(sin^(-1)x))" is "`

Options

A. `-(1)/(sqrt(2))`
B. -1
C. non-existent
D. none of these

Answer

A

Explanation

`Let sin^(-1)x=theta.Then,x=sintheta.`
Now, `xto(1)/(sqrt(2))"or " thetato(pi)/(4)." Therefore,"`
`underset(xto(1)/(sqrt(2)))lim(x-cos(sin^(-1)x))/(1-tan(sin^(-1)x))=underset(thetato(pi)/(4))lim(sintheta-costheta)/(1-tantheta)`
`=underset(thetato(pi)/(4))lim((sintheta-costheta))/((costheta-sintheta))costheta`
`=underset(thetato(pi)/(4))lim-costheta=-(1)/(sqrt(2))`