`("lim")_(xvec0)(sin(x^2))/(1n(cos(2x^2-x)))is equal to` 2 (b) `-2` (c) 1 (d) `-1`

Options

A. `e^(a)`
B. `-a`
C. `e^(1-a)`
D. `e^(1+a)`

Answer

B

Explanation

`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`