If `f(x)=0` is a quadratic equation such that `f(-pi)=f(pi)=0` and `f(pi/2)=-(3pi^2)/4,` then `("lim")_(xvecpi)(f(x))/("sin"(sinx)` is equal to 0 (b) `pi` (c) `2pi` (d) none of these

Options

A. `-(3)/(2)pi`
B. `-(1)/(2)`
C. `(2pi)`
D. `(3)/(2)pi`

Answer

C

Explanation

Given `f(x)=x^(2)-pi^(2)`
`underset(xto-pi^+)lim(x^(2)-pi^(2))/(sin(sinx))=underset(hto0)lim((-pi+h)^(2)-pi^(2))/(sin(sin(-pi+h)))`
`=underset(hto0)lim(-2hpi+h^(2))/(-sin(sinh))`
`=underset(hto0)lim(h-2pi)/((-sin(sinh))/(sinh)xx(sinh)/(h))=2pi`