Limit of Nested Square Root Expression at Infinity: How to Solve?

[transgalactic]

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

??

 

[Mark44]

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.

 

[transgalactic]

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

 

[HallsofIvy]

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

??

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]<br /> <br /> Now use the standard &quot;trick&quot; 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:<br /> \frac{\sqrt{1+ \sqrt{1/x}}}{\sqrt{1+ \sqrt{(1/x)+ \sqrt{1/x^2}}}+ 1}<br /> <br /> As x goes to infinity, each of those fractions goes to 0.