I am having trouble proving the following limit. Any help from the experts?
Show that the limit, as x tends towards infinity, of
$\displaystyle \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}$
is 1/2.
Multiply the numerator and denominator by $\displaystyle \sqrt{x + \sqrt{x + \sqrt{x}}} + \sqrt{x}$. Simplify the resulting expression. Then divide the numerator and denominator by $\displaystyle \sqrt{x}$. Now take the limit.
Edit: I suppose I should have mentioned that the denominator of the original expression is 1 .....