If `lim_(xrarr0) (cosx+a sinbx)^(1//x)= e^2`, then the values of a and b are

Options

A. `a=1,b=-2`
B. `a=2sqrt(2),b=sqrt(2)`
C. `a=2sqrt(2),b=(1)/(xqrt(2))`
D. `a=-2,b=1`

Answer

C

Explanation

We have,
`lim_(xto0) (cosx+a sinbx)^(1//x)= e^2`
` rArr lim_(xto0) [1+(cosx-1+asinbx )]^(1//x)=e^2`
` rArr e^(lim_(xto0) (cosx-1+asin bx)/(x)=e^2`
`rArr e_(lim_(cosx-1)/(x)+ab(sinbx)/(bx) =e^2`
`rArr e^ab=e^2rArr ab =2`
Clearly, `a=2sqrt(2) ` and `b =(1)/(sqrt(2))` satisfy this relation.