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Let $ f(x)= \left\{ \begin{array}{1 1} |x|\cos \frac{1}{x} & \quad if\;x\neq0 \\ 0 & \quad if\;x=0 \end{array} \right. $ then discuss the continuity of f(x) at x = 0

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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} |x|\: cos\large\frac{1}{x}$
$ cos\large\frac{1}{x} = K \: where \: -1 \leq K \leq 1$ for any $ x \neq 0$
 
$ \Rightarrow \lim\limits_{x \to 0} \: K| x |=0$
$ f(0)=0 \Rightarrow f$ is continuous at x = 0

 

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

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