`lim_(xtooo) (cot^(-1)(x^(-a)log_(a)x))/(sec^(-1)(a^(x)log_(x)a)),(agt1),` is equal to

Options

A. `underset(xto1)limf(x)" exists "impliesa=-2`
B. `underset(xto-2)f(x)" exists "impliesa=13`
C. `underset(xto1)f(x)=4//3` if it exists
D. `underset(xto-2)limf(x)=-1//3` if it exists

Answer

B

Explanation

`underset(xtooo)lim(cot^(-1)(x^(-a)log_(a)x))/(sec^(-1)(a^(x)log_(x)a))" "(agt1)`
`=underset(xtooo)lim(cot^(-1)((log_(a)x)/(x^(a))))/(sec^(-1)(a^(x)/(log_(a)x)))`
as` xtooo,((log_(a)x)/(x^(a)))to0" and "(a^(x)/(log_(a)x))tooo`
(using L'Hospital rule)
`:.I=(pi//2)/(pi//2)=1`