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 \( \dfrac12\)?

\[ \lim_{x\to\infty}\left(\sqrt{x+ \sqrt{x +\sqrt{x +\sqrt{x}}}} - \sqrt{x}\right) \]

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