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 \( \frac12?\)

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

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