<s>[INST] limit $\lim_{x \to \infty} \frac{\sqrt{x+2} - \sqrt{x+1}}{\sqrt{x+1} - \sqrt{x}}$ 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.

$$\lim_{x \to \infty} \frac{\sqrt{x+2} - \sqrt{x+1}}{\sqrt{x+1} - \sqrt{x}} $$

 [/INST] \begin{gather*}
\lim _{x\rightarrow \infty }\frac{\sqrt{x+2} -\sqrt{x+1}}{\sqrt{x+1} -\sqrt{x}}\\
=\lim _{x\rightarrow \infty }\frac{\sqrt{1+\frac{2}{x}} -\sqrt{1+\frac{1}{x}}}{\sqrt{1+\frac{1}{x}} -1}\\
\end{gather*}
(Taking $\sqrt{x}$ common from both numerator and denominator)
Now, from the binomial series expansion, $(1+ay)^{b} =1+aby$, when $y\to 0$
\begin{gather*}
\sqrt{1+\frac{2}{x}} =1+\frac{1}{2} \times \frac{2}{x}\\
\sqrt{1+\frac{1}{x}} =1+\frac{1}{2} \times \frac{1}{x}\\
\lim _{x\rightarrow \infty }\frac{\sqrt{1+\frac{2}{x}} -\sqrt{1+\frac{1}{x}}}{\sqrt{1+\frac{1}{x}} -1} =\lim _{x\rightarrow \infty }\frac{\left( 1+\frac{1}{2} \times \frac{2}{x}\right) -\left( 1+\frac{1}{2} \times \frac{1}{x}\right)}{\left( 1+\frac{1}{2} \times \frac{1}{x}\right) -1} =1
\end{gather*}
</s>