If f(x) = |x - 2| write whether f(2) exists or not. - Vidyadip

Question

If f(x) = |x - 2| write whether f(2) exists or not.

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Answer

Given: $\text{f(x)}=|\text{x}-2|=\begin{cases}\text{x}-2, & \text{x}> 2\\-\text{x}+2, & \text{x}\leq 2\end{cases}$
Now,
(LHL at x = 2)
$\lim_\limits{\text{x}\rightarrow2^{-}}\frac{\text{f(x)}-\text{f}(2)}{\text{x}-2}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(2-\text{h})-\text{f}(2)}{2-\text{h}-2}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{(-2+\text{h}+2)-0}{-\text{h}}$
$=-1$
(RHL at x = 2)
$\lim_\limits{\text{x}\rightarrow2^{+}}\frac{\text{f(x)}-\text{f}(2)}{\text{x}-2}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(2+\text{h})-\text{f}(2)}{2+\text{h}-2}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{2+\text{h}+2-0}{\text{h}}$
$=1$
Thus, (LHL at x = 2) $\neq$ (RHL at x = 2)
Hence, $\lim_\limits{\text{x}\rightarrow2}\frac{\text{f(x)}-\text{f}(2)}{\text{x}-2}=\text{f'}(2)$ does not exist.

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