?Left- and Right-Hand Derivatives The left-hand and right-hand derivatives of f at a are
Chapter 2, Problem 65(choose chapter or problem)
Left- and Right-Hand Derivatives The left-hand and right-hand derivatives of f at a are defined by
\(\begin{aligned}
f^{\prime}-(a) & =\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h} \\
\text { and } \quad f^{\prime}+(a) & =\lim _{h \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h}
\end{aligned}\)
if these limits exist. Then \(f^{\prime}(a)\) exists if and only if these one-sided derivatives exist and are equal.
Let
\(f(x)=\left\{\begin{array}{ll}
0 & \text { if } x \leqslant 0 \\
5-x & \text { if } 0<x<4 \\
\frac{1}{5-x} & \text { if } x \geqslant 4
\end{array}\right.\)
(a) Find \(f^{\prime}-(4)\) and \(f^{\prime}+(4)\).
(b) Sketch the graph of f.
(c) Where is f discontinuous?
(d) Where is f not differentiable?
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