Limits to Infinity: Solving for $\frac{2x}{\sqrt{x+2} + \sqrt{x}}$

[PhizKid]

Homework Statement

\lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\

Homework Equations

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?

 

[Curious3141]

Homework Statement

\lim_{x \to \infty} \frac{2x}{\sqrt{x+2} + \sqrt{x}}\\\\\\

Homework Equations

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?

The denominator in the second step is wrong.

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

(for positive x, of course).

 

[PhizKid]

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

 

[chiro]

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