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Home  >>  CBSE XI  >>  Math  >>  Limits and Derivatives
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Evaluate the given limit \[\] $ \lim\limits_{ x \to 0} ( \text{cosec} \: x - \cot\: x)$

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At $ x = 0 $ the value of the given function is $ \infty - \infty$
$ \lim\limits_{ x \to 0} ( \text{cosec} \: x - \cot\: x)$$ = \lim\limits_{ x \to 0}\bigg( \large\frac{1}{ \sin \: x} $$ - \large\frac{ \cos \: x}{\sin \: x}\bigg)$
$ = \lim\limits_{x \to 0} \bigg( \large\frac{1- \cos\: x}{\sin \: x} \bigg)$
$ = \lim\limits_{x \to 0} \large\frac{\bigg( \Large\frac{1- \cos\: x}{x}\bigg) }{\bigg( \Large\frac{\sin \: x}{x}\bigg)} $
$ = \large\frac{\lim\limits_{x \to 0} \Large\frac{1- \cos\: x}{x} }{ \lim\limits_{x \to 0}\Large\frac{\sin \: x}{x}} $
Hint: Using $ \lim\limits_{x \to 0} \large\frac{1- \cos\:x}{x}$$ =0\: and \: \lim\limits_{x \to 0} \large\frac{\sin\: x}{x} $$ =1 $
$ = \large\frac{0}{1} $$= 0$
answered Apr 5, 2014 by thanvigandhi_1
edited May 16, 2014 by balaji.thirumalai
 

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