The function of \(f(x)=|x|+\frac{|x|}{x}\) is
The function of \(f(x)=|x|+\frac{|x|}{x}\) is
- continuous at the origin
- discontinuous at the origin because \(|x|\) is discontinuous there
- discontinuous at the origin because \(\frac{|x|}{x}\) is discontinuous there
- discontinuous at the origin because both \(|x|\) and \(\frac{|x|}{x}\) are discontinuous
Solution
\(f(x)=|x|+\frac{|x|}{x}\) LHL of \(x=0\)
\(\begin{aligned}
\lim _{x \rightarrow 0^{-}} f(x) & =\lim _{x \rightarrow 0^{-}}-x+\frac{-x}{x} \\
& =\lim _{x \rightarrow 0^{-}}-x-1=-1 \\
\lim _{x \rightarrow 0^{+}} f(x) & =\lim _{x \rightarrow 0^{+}}(x)+\frac{(x)}{x} \\
& =\lim _{x \rightarrow 0^{+}} x+1=1
\end{aligned}\)
\(\therefore\) LHL \(\neq\) RHL at \(x=0\)
\(\therefore f(x)\) is discontinuous at \(x=0\)
Hence, option (c) is correct.
Asked in: AP EAMCET 2020 (18 Sep Shift 2)
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