Sqrt limit question

1. Feb 12, 2009

transgalactic

i need to solve this limit
$$\lim_{x->\infty}\left ( \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)$$
i tried
$$\lim_{x->\infty}\left ( \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)=\\ \lim_{x->\infty}\left (\frac{\frac{1}{\sqrt{x}}\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}}{\frac{1}{\sqrt{x}}} \right)$$
but i get 0/0

??

Last edited: Feb 13, 2009
2. Feb 12, 2009

Staff: Mentor

Well, 0/0 is not an answer. Put some numbers in and see what you get. That will at least give you an idea of what the limit might be.

3. Feb 13, 2009

transgalactic

i agree that 0/0 is not an answer
how to solve it?

4. Feb 13, 2009

HallsofIvy

Staff Emeritus
That looks to me like a candidate for "rationalizing" Write it as
$$\frac{\left ( \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}\right)}{1}$$
and multiply both numerator and denominator by
$$\left ( \sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}\right)$$

You will get
$$\frac{x+ \sqrt{x+\sqrt{x}}- x}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}[/itex] Now use the standard "trick" when x is going to infinity: divide both numerator and denominator by the highest power of x, here $\sqrt{x}$, so every x is moved to the denominator: [tex]\frac{\sqrt{1+ \sqrt{1/x}}}{\sqrt{1+ \sqrt{(1/x)+ \sqrt{1/x^2}}}+ 1}$$

As x goes to infinity, each of those fractions goes to 0.