`lim_(xto0) (1)/(x)cos^(1)((1-x^(2))/(1+x^2))` is equal to

Options

A. `f(x)f(y)`
B. `f(x)+f(y)`
C. `f(x)-f(y)`
D. none of these

Answer

D

Explanation

We know that `cos^(-1)((1-x^(2))/(1+x^2))={{:(2tan^(-1)x", "xge0),(-2tan^(-1)x", "xle0):}`
or `underset(xto0)lim(1)/(x)cos^(-1)((1-x^(2))/(1+x^2))=underset(xto0^(+))lim(2tan^(-1)x)/(x)=2`
and`underset(xto0)lim(1)/(x)cos^(-1)((1-x^(2))/(1+x^2))=underset(xto0^(-))lim[-(2tan^(-1)x)/(x)]=-2`