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Compute the limit of the given function

Evaluate the limit.

    \[ \lim_{x \to +\infty} x^{\frac{1}{4}} \sin \left( \frac{1}{\sqrt{x}} \right). \]


First, we make the substitution t = \frac{1}{\sqrt{x}}. Then t \to 0^+ as x \to +\infty so we have

    \begin{align*}  \lim_{x \to +\infty} x^{\frac{1}{4}} \sin \left( \frac{1}{\sqrt{x}} \right) &= \lim_{t \to 0^+} \frac{\sin t}{\sqrt{t}} \\[9pt]  &= \lim_{t \to 0^+} \frac{\cos t}{\frac{1}{2\sqrt{t}}} &(\text{L'Hopital's})\\[9pt]  &= \lim_{t \to 0^+} 2 \sqrt{t} \cos t \\[9pt]  &= 0. \end{align*}

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