Conjugation on limit
Calculus
Level
1
\[ \large \lim_{x\to\infty}\left( \sqrt{x +\sqrt{x}} - \sqrt{x}\right), \quad \lim_{x\to\infty}\left( \sqrt{x +\sqrt{x +\sqrt{x}}} - \sqrt{x}\right) \]
It is possible to show that both the limits above are equal to \( \frac12.\)
Is it true that the limit below is also equal to \( \frac12?\)
\[ \lim_{x\to\infty}\left(\sqrt{x+ \sqrt{x +\sqrt{x +\sqrt{x}}}} - \sqrt{x}\right) \]
by
Naren Bhandari