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Question
Write the derivative of f (x) = |x|3 at x = 0.
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Solution
Given:
(LHD at x = 0)
\[\lim_{x \to 0^-} \frac{f(x) - f(0)}{x - 0}\]
\[ = \lim_{h \to 0} \frac{f(0 - h) - f(0)}{x}\]
\[ = \lim_{h \to 0} \frac{h^3}{- h} \]
\[ = 0\]
(RHD at x = 0)
\[\lim_{x \to 0^+} \frac{f(x) - f(0)}{x - 0} \]
\[ = \lim_{x \to 0^+} \frac{f(0 + h) - f(0)}{x}\]
\[ = \lim_{h \to 0} \frac{h^3 - 0}{h} \]
\[ = 0\]
and
Thus, (LHD at x=0) = (RHD at x = 0) =
Hence,
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