Limit of $h(x)$ as $x \to \infty$ Let $h(x) = \sqrt{x + \sqrt{x}} - \sqrt{x}$ , find $\lim_{x \to \infty } h(x)$ . I've tried substitution , multiplying by conjugate and l'hospital's rule but didn't work .  

Multiplying and dividing by the "conjugate" $\sqrt{x + \sqrt{x}} + \sqrt{x}$ is a good strategy! 
Note that for $x>0$,
$$ \sqrt{x + \sqrt{x}} - \sqrt{x}=\frac{ (x + \sqrt{x}) - x}{ \sqrt{x + \sqrt{x}} + \sqrt{x}}=\frac{\sqrt{x}}{\sqrt{x} \sqrt{1 + \frac{1}{\sqrt{x}}} + \sqrt{x}}=\frac{ 1}{ \sqrt{1 + \frac{1}{\sqrt{x}}} + 1}.$$
Can you take it from here?
