Simple Limits Problem: Finding the Limit of a Square Root Expression

[Saitama]

Homework Statement

Find
$$\lim_{x\rightarrow \infty} (\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})$$

Homework Equations

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!

 

[mfb]

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.

 

[Curious3141]

Homework Statement

Find
$$\lim_{x\rightarrow \infty} (\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x})$$

Homework Equations

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!

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.

 

[Saitama]

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.

Thanks mfb!