Answer
The given limit is $f'(1)$ where $f(x)=\sqrt x$.
Work Step by Step
By definition, the derivative at a point $a$ is
$f'(a)=\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}=\lim\limits_{h \to 0}\frac{\sqrt {1+h}-1}{h}$
$\implies f(a+h)=\sqrt {1+h}$ and $f(a)=1$
$\implies a=1$ and $f(x)=\sqrt x$
The given limit is $f'(1)$ where $f(x)=\sqrt x$
