Evaluate the following one sided limits: _ x - 2^+ - Vidyadip

Question

Evaluate the following one sided limits: $\lim\limits_{\text{x}\rightarrow\frac{-\pi}{2^+}}\sec\text{x}$

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Answer

$\lim\limits_{\text{x}\rightarrow\frac{\pi^+}{2}}\sec\text{x}$ $=\lim\limits_{\text{h}\rightarrow0}\sec\big(-\frac{\pi}{2}+\text{h}\big)$ $=\sec\Big(-\frac{\pi}{2}+0\Big)$ $=\sec\Big(-\frac{\pi}{2}\Big)$ $=\frac{1}{\cos\Big(-\frac\pi2\Big)}$ $=\frac{-1}{\Big(\cos\frac{\pi}{2}\Big)}$ $\Rightarrow\frac{-1}{0}=-\infty$

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