Answer
$ \dfrac{1}{2}$
Work Step by Step
In order to to find the solution we will rationalize the function and then simplify.
$\dfrac{\sqrt {1+x}-1}{x} \times \dfrac{\sqrt {1+x}+1}{\sqrt {1+x}+1}$
or, $=\dfrac{(1+x)-1}{x[\sqrt {1+x}+1]}$
or, $=\dfrac{1}{\sqrt {1+x}+1}$
Now, $\lim_\limits{x\to 0} \dfrac{\sqrt {1+x}-1}{x}=\lim_\limits{x\to 0} \dfrac{1}{\sqrt {1+x}+1}$
or, $= \dfrac{1}{\sqrt {1+0}+1}$
or, $= \dfrac{1}{2}$
