Answer
$\frac{\sqrt{x}-1}{x-1}$
Work Step by Step
We simplify the fraction by multiplying through by $\sqrt{x}-1$ and using the fact that $(a-b)(a+b)=a^2-b^2$:
$\displaystyle \frac{1}{\sqrt{x}+1}=\frac{1}{\sqrt{x}+1} \displaystyle *\frac{\sqrt{x}-1}{\sqrt{x}-1}=\frac{\sqrt{x}-1}{x-1}$
