Evaluate the following limit: `lim_(x->0)(cos x+a sin b x)^(1//x)`

Explanation

`lim_(x->0)(cos x+a sin b x)^(1//x)`

`=lim_(x->0)(cos x+a sin b x -1)^(1//x)`

As we know that , if `lim(x to a) f(x)=lim_(x to a) g(x)=0` such that `lim_(x to a) (f(x))/(g(x))` exists, then `lim_(x to a)(1+f(x))^(1/(g(x)))=e^(lim_(x to a) (f(x))/(g(x))`

`=e^(lim_(x->0)[(cos x+a sin b x-1)/x])`

`=e^(lim_(x->0)[(b xx asinbx)/(bx)-((1-cosx))/(x)]`

`=e^(lim_(x->0) [(ab sin bx)/(bx)-
(2sin^2(x/2))/(x)]`

`=e^(ab)`