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\( \lim _{x \rightarrow 0} \frac{\operatorname{cosec} x-\cot x}{x} \) is |
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Answer» \(\lim\limits_{x\to 0}\frac{cosecx-cot x}x\) = \(\lim\limits_{x\to 0}\frac{1-cos x}{xsin x}\) [0/0 case] = \(\lim\limits_{x\to 0}\frac{sin x}{xcos x+sin x}\) [By using D.L.H. Rule] = \(\lim\limits_{x\to 0}\cfrac{x\frac{sin x}x}{x(cos x+\frac{sin x}x)}\) = \(\lim\limits_{x\to 0}\cfrac{\lim\limits_{x\to 0}\frac{sin x}x}{cos x+\lim\limits_{x\to 0}\frac{sin x}x}\) = \(\lim\limits_{x\to 0}\) \(\frac1{cos x+1}\) (\(\because\) \(\lim\limits_{x\to 0}\frac{sin x}x=1\)) = \(\frac1{1+cos 0}=\frac1{1+1} = \frac12\) |
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