Evaluate `lim_(x to 0) (2^(x)-1)/((1+x)^(1//2)-1)`

Options

A. `(1)/(8sqrt(3))`
B. `(1)/(4sqrt(3))`
C. 0
D. none of these

Answer

`log4`

Explanation

`underset(xto0)lim(2^(x)-1)/((1+x)^(1//2)-1)`
`=underset(xto0)lim(2^(x)log2)/((1)/(2)(1+x)^(-1//2))" "`(Applying L'Hospital's rule)
`=2log2`
`=log4`