I can't solve this limit...

Question

Limit

I tried to solve it as difference of two squares. But I guess I can't move any longer from that. Please help...

score 2 · Accepted Answer · answered Oct 04 '13 at 18:43

2

Rationalize the numerator to get $$\lim_{x \to \infty} \frac{ \sqrt{x + \sqrt{x + \sqrt{x}}}}{\sqrt{ x + \sqrt{x + \sqrt{x + \sqrt{x}}}} + \sqrt{x}}$$ You can eliminate the indeterminacy by multiplying top and bottom by $\frac{1}{\sqrt{x}}$.

answered Oct 04 '13 at 18:43

Umberto P.

52,165

Ahaha, I wish I didn't ask this :) – A6SE Oct 04 '13 at 18:49

score 2 · Answer 2 · answered Oct 04 '13 at 18:51

Factorizing by $\sqrt{x}$, this quantity becomes \begin{align} \sqrt{x}\times \left(\sqrt{1+\frac{\sqrt{x + \sqrt{x + \sqrt{x}}}}{x}}-1\right) &\sim \sqrt{x}\times \frac{1}{2}\frac{\sqrt{x + \sqrt{x + \sqrt{x}}}}{x}\\ & \sim \frac{1}{2}\sqrt{1+\frac{\sqrt{x+\sqrt{x}}}{x}} \end{align} So the limit is $\dfrac{1}{2}$.

I can't solve this limit...

2 Answers2