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Find the limit as x goes to (π / 2) of cos x / (x – (π / 2))

by RoRi

Evaluate the limit.

    \[ \lim_{x \to \frac{\pi}{2}} \frac{\cos x}{x - \frac{1}{2} \pi}. \]

Since \cos x = -\sin \left( x - \frac{\pi}{2} \right) we use the expansion (page 287) for \sin x,

    \[ \sin \left( x - \frac{\pi}{2} \right) = \left( x - \frac{\pi}{2} \right) + o \left( \left( x - \frac{\pi}{2} \right)^2 \right). \]

Therefore,

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

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