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.

Printable View

- Aug 11th 2008, 11:11 PMtombrowningtondifficult limit
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. - Aug 11th 2008, 11:19 PMmr fantastic
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 .....