Help calculating $\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$ I need some help calculating this limit:
$$\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$$
I know it's equal to 1 but I have no idea how to get there. Can anyone give me a tip? I can't use l'Hopital. Thanks a lot.

By Lagrange's theorem, $a>b>0$ ensures $\sqrt{a}-\sqrt{b} = (a-b)\frac{1}{2\sqrt{c}}$ with $c\in(b,a)$.
If we let $a=x+\sqrt{x}$ and $b=x-\sqrt{x}$ we get
$$ \sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}} = \frac{2\sqrt{x}}{2\sqrt{c}},\quad c\in(x-\sqrt{x},x+\sqrt{x})$$
and since $\sqrt{x\pm\sqrt{x}}=\sqrt{x}(1+o(1))$ the outcome is clear.
