`underset(xto0)lim(sin(picos^(2)x))/(x^(2))` is equal to

Options

A. 1
B. `-1pi`
C. `pi`
D. `pi/2`

Answer

C

Explanation

`lim_(x to 0) (sin(pi cos^(2)x))/(x^(2)) = lim_(x to 0) (sin {pi-pi (cos^(2)x)})/(x^(2))`
`=lim_( x to 0) (sin (pi sin^(2)x))/(x^(2)) = lim_(x to 0) (sin (pi sin^(2)x))/((pi sin^(2)x))*(pisin^(2)x)/(x^(2))`
`=lim_(x to 0) (sin(pi sin^(2)x))/(pi sin^(2)x)*pi*((sinx)/(x))^(2)=1*pi*1=pi`