$\text {If } \begin{aligned} f(x) &=\frac{|x-2|}{x-2}, \quad \text { for } x \neq 2 \\ &=1 \quad, \quad…

$\text {If } \begin{aligned} f(x) &=\frac{|x-2|}{x-2}, \quad \text { for } x \neq 2 \\ &=1 \quad, \quad \text { for } x=2, \end{aligned}$ then which of the following statements is true?
  1. $f(x)$ is continuous at $x=2$
  2. $\lim _{x \rightarrow 2^{-}} f(x)=f(2)$
  3. $\lim _{x \rightarrow 2^{+}} f(x)=\lim _{x \rightarrow 2^{-}} f(x)$
  4. $f(x)$ is discontinuous at $x=2$

Solution

Here $\begin{aligned}|x-2| &=x-2, \quad \text { if } x>2 \Rightarrow \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=0 \\ &=-(x-2), \text { if } x < 2 \end{aligned}$ But $f(2)=1 \neq 0$ So $f(x)$ is discontinuous at $x=2$

Asked in: MHT CET 2020 (15 Oct Shift 1)

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