The value of `lim_(xrarr0^(+))((1-cos(sin^(2)x))/(x^(2)))^((log_(e)(1-2x^(2)))/(sin^(2)x))` is

Options

A. 0
B. e
C. `-1`
D. `oo`

Answer

D

Explanation

`underset(xrarr0^(+))(lim)((1-cos(sin^(2)x))/(x^(2)))^((log_(e)(1-2x^(2)))/(sin^(2)x))`
As `underset(xrarr0^(+))(lim)((1-cos(sin^(2)x))/(sin^(4)x))xx(sin^(4)x)/(x^(2))=(1)/(2)xx=0`
`and underset(xrarr0^(+))(lim)((ln(1-2x^(2))/(-2x^(2))).(-2x^(2)))/(((sin^(2)x)/(x^(2))).x^(2))=-2`
So, `underset(xrarr0^(+))(lim)((1-cos(sin^(2)x))/(x^(2)))^((ln(1-2x^(2)))/(sin^(2)x))=oo`