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("lim")(x vec 0)(sin(x^2))/(1n(cos(2x^2-...

`("lim")_(x vec 0)(sin(x^2))/(1n(cos(2x^2-x)))` is equal to (a) `2 `(b) `-2` (c) `1` (d) `-1`

A

`e^(a)`

B

`-a`

C

`e^(1-a)`

D

`e^(1+a)`

Text Solution

Verified by Experts

The correct Answer is:
B
`underset(xto0)lim(sin(x^(2)))/("ln"(cos(2x^(2)-x)))`
`=underset(xto0)lim(sin(x^(2)))/(log(1-2sin^(2)((2x^(2)-x)/(2))))`
`=underset(xto0)lim(sin(x^(2))x^(2))/((x^(2)log(1-2sin^(2)((2x^(2)-x)/(2))))/(-2sin^(2)((2x^(2)-x)/(2)))[-2sin^(2)((2x^(2)-x)/(2))])`
`=underset(xto0)lim-(x^(2))/((2sin^(2)((2x^(2)-x)/(2)))/(((2x^(2)-x)/(2))^(2))((2x^(2)-x)/(2))^(2))`
`=underset(xto0)lim-(2x^(2))/((2x^(2)-x)^(2))=underset(xto0)lim-(2)/((2x-1)^(2))=-2`
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