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The value of lim(xrarr0^(+))((1-cos(sin^...

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

A

0

B

e

C

`-1`

D

`oo`

Text Solution

Verified by Experts

The correct Answer is:
D
`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`
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