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Find the limit as x goes to 1 of given function

by RoRi

Evaluate the limit.

    \[ \lim_{x \to 1} \frac{\sin \left( \frac{\pi}{2x} \right) \cdot \log x}{(x^3+5)(x-1)}. \]

Using the expansion (from this exercise, Section 7.11, Exercise #4)

    \[ \log x = (x-1) + o(x-1) \]

we compute the limit as follows:

    \begin{align*} \lim_{x \to 1} \frac{\sin \left( \frac{\pi}{2x} \right) \cdot \log x}{(x^3 + 5)(x-1)} &= \lim_{x \to 1} \left( \frac{\sin \left( \frac{\pi}{2x} \right)}{x^3+5} \right) \left(\frac{\log x}{x-1} \right) \\[9pt] &= \frac{1}{6} \lim_{x \to 1} \frac{\log x}{x-1} \\[9pt] &= \frac{1}{6} \lim_{x \to 1} \frac{(x-1) + o(x-1)}{x-1} \\[9pt] &= \frac{1}{6}. \end{align*}

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