# Simple limits problem

1. Jul 9, 2013

### Pranav-Arora

1. The problem statement, all variables and given/known data
Find
$$\lim_{x\rightarrow \infty} (\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})$$

2. Relevant equations

3. The attempt at a solution
Rewriting the given expression,
$$\sqrt{x}\left(\sqrt{1+\sqrt{\frac{1}{x}\left(1+\frac{1}{\sqrt{x}}\right)}}-1\right)$$
What should I do with the sqrt(x) outside?

Any help is appreciated. Thanks!

2. Jul 9, 2013

### Staff: Mentor

A common way to approach limits of the type of $\sqrt{x+f(x)}-\sqrt{x}$ is a multiplication with $\displaystyle 1=\frac{\sqrt{x+f(x)}+\sqrt{x}}{\sqrt{x+f(x)}+\sqrt{x}}$. This does not change the limit (as you multiply with 1), but you can simplify the numerator a lot.

3. Jul 9, 2013

### Curious3141

I wanted to post earlier but kept messing up my algebra.

Call the expression $y$. Find $y.(\sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt x)$.

mfb has suggested pretty much the same thing.

4. Jul 9, 2013

Thanks mfb!