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Question
Show that f(x) = |x − 2| is continuous but not differentiable at x = 2.
Solution
Given:
Continuity at x=2: We have,
(LHL at x = 2)
\[{= \lim}_{x \to 2^-} f(x) \]
\[ = \lim_{h \to 0} f(2 - h) \]
\[ = \lim_{h \to 0} ( - 2 + h) + 2\]
\[ = 0\]
(RHL at x = 2)
\[{= \lim}_{x \to 2^+} f(x) \]
\[ = \lim_{h \to 0} f(2 + h) \]
\[ = \lim_{h \to 0} 2 + h - 2 \]
\[ = 0\]
and
Hence,
\[ = \lim_{x \to 2} \frac{( - x + 2) - 0}{x - 2} \]
\[ = \lim_{x \to 2} \frac{- (x - 2)}{x - 2} \]
\[ = \lim_{x \to 2} ( - 1) \]
\[ = - 1\]
=
\[ = \lim_{x \to 2} \frac{(x - 2) - 0}{x - 2} \]
\[ = \lim_{x \to 2} 1 \]
\[ = 1\]
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