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

I don't understand where I did my work wrong. I divided all the terms by a common multiple to simplify them, so I don't see anywhere I could have done it incorrectly.