How can I find the limit to infinity of this function? As this is a $0/0$ equation, I tried using the L'Hôpital's rule in this but ended up making it more complex. I've also tried rationalising the denominator but it didn't lead to anywhere.
$$\lim_{x \to \infty} \frac{\sqrt{x+2} – \sqrt{x+1}}{\sqrt{x+1} – \sqrt{x}} $$
Best Answer
Assume WLOG $x\geq0$. Thus, $$\frac{\sqrt{x+2}-\sqrt{x+1}}{\sqrt{x+1}-\sqrt x}\cdot\frac{\sqrt{x+1}+\sqrt x}{\sqrt{x+1}+\sqrt x}$$$$=(\sqrt{x+1}+\sqrt x)(\sqrt{x+2}-\sqrt{x+1})\cdot\frac{\sqrt{x+2}+\sqrt{x+1}}{\sqrt{x+2}+\sqrt{x+1}}$$$$=\frac{\sqrt{x+1}+\sqrt x}{\sqrt{x+2}+\sqrt{x+1}}=1-\frac{\sqrt{x+2}-\sqrt x}{\sqrt{x+2}+\sqrt{x+1}}$$