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Chapter 3 | Derivatives
Figure 3.14 The function f ( x ) = | x | is continuous at 0 but is not differentiable at 0.
Let’s consider some additional situations in which a continuous function fails to be differentiable. Consider the function f ( x ) = x 3 : 1 x 3 2 =+∞. Thus f ′(0) does not exist. A quick look at the graph of f ( x ) = x 3 clarifies the situation. The function has a vertical tangent line at 0 ( Figure 3.15 ). f ′(0) = lim x →0 x 3 −0 x −0 = lim x →0
Figure 3.15 The function f ( x ) = x 3 has a vertical tangent at x =0. It is continuous at 0 but is not differentiable at 0.
⎧ ⎩ ⎨ x sin
⎛ ⎝ 1 x
⎞ ⎠ if x ≠0
The function f ( x ) =
also has a derivative that exhibits interesting behavior at 0. We see that
0 if x =0
f ′(0) = lim x →0 ⎞ ⎠ . This limit does not exist, essentially because the slopes of the secant lines continuously change direction as they approach zero ( Figure 3.16 ). x sin(1/ x )−0 x −0 = lim x →0 sin ⎛ ⎝ 1 x
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