lim-x-x-x-x-x-x-please-solution-

Question Number 186500 by 073 last updated on 05/Feb/23

\lim_{x \to + \infty} \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x}}}} - \sqrt{x} = ?

please solution

Answered by greougoury555 last updated on 05/Feb/23

\lim_{x \to \infty} \frac{\sqrt{x + \sqrt{x + \sqrt{x}}}}{\sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x}}}} + \sqrt{x}}

= \lim_{x \to \infty} \frac{\sqrt{x} (\sqrt{1 + \sqrt{\frac{1}{x} + \sqrt{\frac{1}{x}}}})}{\sqrt{x} (\sqrt{1 + \sqrt{\frac{1}{x} + \sqrt{\frac{1}{x^{2}} + \sqrt{\frac{1}{x^{4}}}}}} + 1)}

= \frac{1}{2}