`lim_(x to (pi)/(2))(a^(cot x) -a^(cosx))/(cot x- cot x )` is equalt o

Options

A. `log a`
B. `log 2`
C. a
D. log x

Answer

A

Explanation

`underset(x to (x)/(2))(lim)((a^(cotx)-a^(cos x))/(cot x - cot x))`
`=underset(x to (pi)/(2))(lim) a^(cosx)((a^(cot x - cos x) -1)/(cotx - cosx))`
`=a^(cos.(pi)/(2))underset(x to (pi)/(2))(lim)(because underset(x to0)(lim) (a^(x)-1)/(x) = log a)`
`= 1 log a = loga( because underset(x to0)(lim) (a^(x)-1)/(x) = log a)`