Conjugation on limit
Calculus
Level
3
\[ \large \lim_{x\to\infty}\left( \sqrt{x +\sqrt{x}} - \sqrt{x}\right) \qquad , \qquad \lim_{x\to\infty}\left( \sqrt{x +\sqrt{x +\sqrt{x}}} - \sqrt{x}\right) \]
It is possible to show that both the limits above are equal to \( \dfrac12\).
Is it also true that the following limit is also equal to \( \dfrac12\)?
\[ \lim_{x\to\infty}\left(\sqrt{x+ \sqrt{x +\sqrt{x +\sqrt{x}}}} - \sqrt{x}\right) \]
by
Naren Bhandari