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Compute the limit of the given function

Evaluate the limit.

    \[ \lim_{x \to +\infty} \frac{\sin \frac{1}{x}}{\arctan \frac{1}{x}}. \]


Let t = \frac{1}{x}, then t \to 0 as x \to +\infty. Making this substitution and using L’Hopital’s rule we have

    \begin{align*}  \lim_{x \to +\infty} \frac{\sin \frac{1}{x}}{\arctan \frac{1}{x}} &= \lim_{t \to 0} \frac{\sin t}{\arctan t} \\[9pt]  &= \lim_{t \to 0} \frac{\cos t}{\frac{1}{1+t^2}} \\[9pt]  &= 1.  \end{align*}

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