Evaluate `lim_(hto0)(sqrt(x+h)-sqrt(x))/(h)`

Options

A. `1/(2sqrt(x))`
B. `1/(sqrt(x))`
C. `1/(2x)`
D. none of these

Answer

A

Explanation

Given, `underset(hto0)"lim"(sqrt(x+h)-sqrt(x))/(h) = underset(hto0)"lim"((x+h)^(1//2)-(x)^(1//2))/(x+h-x)`
`=underset(hto0)"lim"((x+h)^(1//2)-(x)^(1//2))/((x+h)-x)` `[therefore underset(xtoa)"lim"(x^(n)-a^(n))/(x-a) = na^(n-1)]`
`=1/2x^(1/2-1) = 1/2x^(-1//2)` `[therefore h-0 rArr x+h to x]`
`=1/(2sqrt(x))`