Answer
The solution is
$$\lim_{x\to\infty}(\sqrt{x-2}-\sqrt{x-4})=0.$$
Work Step by Step
To solve this limit follow the steps below
$$\lim_{x\to\infty} (\sqrt{x-2}-\sqrt{x-4})=\lim_{x\to\infty}(\sqrt{x-2}-\sqrt{x-4})\cdot\frac{\sqrt{x-2}+\sqrt{x-4}}{\sqrt{x-2}+\sqrt{x-4}}=\lim_{x\to\infty}\frac{\sqrt{x-2}^2-\sqrt{x-4}^2}{\sqrt{x-2}+\sqrt{x-4}}=\lim_{x\to\infty}\frac{x-2-(x-4)}{\sqrt{x-2}+\sqrt{x-4}}=\lim_{x\to\infty}\frac{2}{\sqrt{x-2}+\sqrt{x-4}}=\left[\frac{2}{\sqrt{\infty-2}+\sqrt{\infty-4}}\right]=\left[\frac{2}{\infty}\right]=0.$$
