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Question: Answered & Verified by Expert
If $f(x)=|x-2|$, then
MathematicsContinuity and DifferentiabilityJEE Main
Options:
  • A $\lim _{x \rightarrow 2}^{+} f(x) \neq 0$
  • B $\lim _{x \rightarrow 2}^{-} f(x) \neq 0$
  • C $\lim _{x \rightarrow 2}^{+} f(x) \neq \lim _{x \rightarrow 2-} f(x)$
  • D $f(x)$ is continuous at $x=2$
Solution:
2208 Upvotes Verified Answer
The correct answer is: $f(x)$ is continuous at $x=2$
$\begin{aligned} & \text {Here } f(2)=0 \\ & \lim _{x \rightarrow 2}^{-} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}|2-h-2|=0 \\ & \lim _{x \rightarrow 2}^{+} f(x)=\lim _{h \rightarrow 0} f(2+h)=\lim _{h \rightarrow 0}|2+h-2|=0\end{aligned}$
Hence it is continuous at $x=2$.

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