Question

Discuss the continuity of the following functions at the indicated point(s): 

\[f\left( x \right) = \left\{ \begin{array}{l}\frac{2\left| x \right| + x^2}{x}, & x \neq 0 \\ 0 , & x = 0\end{array}at x = 0 \right.\]

Answer 1

Given:

\[f\left( x \right) = \binom{\frac{2\left| x \right| + x^2}{x}, x \neq 0}{0, x = 0}\] 

\[\Rightarrow f\left( x \right) = \begin{cases}\frac{2x + x^2}{x}, x > 0 \\ \frac{- 2x + x^2}{x}, x < 0 \\ 0, x = 0\end{cases}\] 

\[\Rightarrow f\left( x \right) = \begin{cases}\left( x + 2 \right), x > 0 \\ \left( x - 2 \right), x < 0 \\ 0, x = 0\end{cases}\] 

We observe

(LHL at x = 0) = 

\[\lim_{x \to 0^-} f\left( x \right) = \lim_{h \to 0} f\left( - h \right) = \lim_{h \to 0} \left[ - h - 2 \right] = - 2\]

(RHL at x = 0) = \[\lim_{x \to 0^+} f\left( x \right) = \lim_{h \to 0} f\left( h \right) = \lim_{h \to 0} \left( 2 + h \right) = 2\]

\[\Rightarrow \lim_{x \to 0^-} f\left( x \right) \neq \lim_{x \to 0^+} f\left( x \right)\]

Hence, f(x) is discontinuous at x = 0.