Evaluate the following one sided limits: _ x 0^- (1+cosec x) - Vidyadip

Question

Evaluate the following one sided limits: $\lim\limits_{\text{x}\rightarrow0^-}(1+\text{cosec }\text{x})$

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Answer

$\lim\limits_{\text{x}\rightarrow0^-}(1+\text{cosec }\text{x})$ $=\lim\limits_{\text{x}\rightarrow0^-}1+\text{cosec }(0-\text{h})$ $=\lim\limits_{\text{h}\rightarrow0}\ 1-\text{cosec}\text{ h}$ $=\lim\limits_{\text{h}\rightarrow0}1-\frac{1}{\sin\text{h}}$ $\Rightarrow1-\frac{1}{0}=-\infty$

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