Limits to infinity

1. Oct 27, 2012

PhizKid

1. The problem statement, all variables and given/known data
$$\lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\$$

2. Relevant equations

3. The attempt at a solution
$$\lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\ \lim_{x \to \infty} \frac{\frac{2x}{x}}{\sqrt{\frac{x}{x}+\frac{2}{x}} + \sqrt{\frac{x}{x}}}\\\\\\ \lim_{x \to \infty} \frac{2}{\sqrt{1 + \frac{2}{x}} + \sqrt{1}}\\\\\\ \lim_{x \to \infty} \frac{2}{\sqrt{1} + \sqrt{1}}\\\\\\ \lim_{x \to \infty} \frac{2}{1 + 1} = \frac{2}{2} = 1\\\\\\$$

But this is incorrect. Where have I done my work incorrectly?

2. Oct 27, 2012

Curious3141

The denominator in the second step is wrong.

$$\frac{\sqrt{f(x)}}{x} = \sqrt{\frac{f(x)}{x^2}}$$

(for positive x, of course).

3. Oct 27, 2012

PhizKid

I don't understand why you have to divide by x^2 and not just x.

4. Oct 27, 2012

chiro

Its because SQRT(x^2) = x if x >= 0.