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Question
Write the value of \[\lim_{x \to \pi/2} \frac{2x - \pi}{\cos x} .\]
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Solution
\[\lim_{x \to \frac{\pi}{2}} \left[ \frac{2x - \pi}{\cos x} \right]\]
\[LHL: \]
\[ \lim_{x \to \frac{\pi}{2}^-} \left[ \frac{2x - \pi}{\cos x} \right]\]
\[Let x = \frac{\pi}{2} - h\]
\[\text{ If } x \to \frac{\pi}{2}, \text{ then we have }: \]
\[ h \to 0\]
\[ = \lim_{h \to 0} \left[ \frac{2\left( \frac{\pi}{2} - h \right) - \pi}{\cos \left( \frac{\pi}{2} - h \right)} \right]\]
\[ = \lim_{h \to 0} \left[ \frac{\pi - 2h - \pi}{\sin h} \right]\]
\[ = - 2\]
\[RHL: \]
\[ \lim_{x \to \frac{\pi}{2}^+} \left[ \frac{2x - \pi}{\cos x} \right]\]
\[\text{ Let } x = \frac{\pi}{2} + h\]
\[\text{ If } x \to \frac{\pi}{2}, \text{ then we have }: \]
\[h \to 0\]
\[ = \lim_{h \to 0} \left[ \frac{2\left( \frac{\pi}{2} + h \right) - \pi}{\cos \left( \frac{\pi}{2} + h \right)} \right]\]
\[ = \lim_{h \to 0} \left[ \frac{2h}{- \sin h} \right]\]
\[ = - 2\]
\[ \Rightarrow \lim_{x \to \frac{\pi}{2}} \left( \frac{2x - \pi}{\cos x} \right) = - 2\]
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