Evaluate the following limits in Exercise: _ x 0 (cosec x- ) - Vidyadip

Question

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow0}(\text{cosec x}-\cot\text{x})$

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Answer

At x = 0, the value of the given function takes the form$\infty-\infty$. Now, $\lim\limits_{\text{x}\rightarrow0}(\text{cosec x}-\cot\text{x})$ $=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}\Big)$ $=\lim\limits_{\text{x}\rightarrow0}\Big(\frac{1-\cos\text{x}}{\sin\text{x}}\Big)$ $=\lim\limits_{\text{x}\rightarrow0}\frac{\Big(\frac{1-\cos\text{x}}{\text{x}}\Big)}{\Big(\frac{\sin\text{x}}{\text{x}}\Big)}$ $=\frac{\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos\text{x}}{\text{x}}}{\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}}$ $=\frac{0}{1}$ $\bigg[{\lim\limits_{\text{x}\rightarrow0}\frac{1-\cos\text{x}}{\text{x}}=0}\text{and}{\lim\limits_{\text{x}\rightarrow0}\frac{\sin\text{x}}{\text{x}}}=1\bigg]$ $=0$

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