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Discuss the continuity of the following function at x = 0 $f(x)= \left\{ \begin{array}{1 1} \Large\frac{x^4+x^3+2x^2}{\tan ^{-1}x} & \quad ,\;x\neq0 \\ 10 & \quad ,\;x=0 \end{array} \right. $

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  • For continuity at '0' LHL = RHL = f(0)
  • $ \lim\limits_{x \to 0} \: \large\frac{tan^{-1}x}{x}=1 $
LHL = RHL
$= \lim\limits_{x \to 0} \: \large\frac{\large x^4+x^3+2x^2}{tan^{-1}x}$
$= \lim\limits_{x \to 0} \: \large\frac{\large x^3+x^2+2x}{\large\frac{tan^{-1}x}{x}} = 0 $
 
$ f(0) = 10$
$ \Rightarrow $ f is not continuous.

 

answered Mar 11, 2013 by thanvigandhi_1
edited Mar 26, 2013 by thanvigandhi_1
 

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