The value of the limit `lim_(xto0) (a^(sqrt(x))-a^(1//sqrt(x)))/(a^(sqrt(x))+a^(1//sqrt(x))),agt1,`is

Options

A. does not exist
B. `1//3`
C. 0
D. `2//9`

Answer

C

Explanation

`underset(xto0)lim(a^(sqrt(x))-a^(1//sqrt(x)))/(a^(sqrt(x))+a^(1//sqrt(x))),agt1`
Put `x=r^(2)."Then"`
`underset(t to0)lim(a^(t)-a^(1//t))/(a^(t)+a^(1//t))`
or`underset(t to0)lim(a^(t-1//t)-1)/(a^(t-1//t)+1)=(a^(-oo)-1)/(a^(-oo)+1)=-1`