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Question
Determine whether the following function is differentiable at the indicated values.
f(x) = sin |x| at x = 0
Solution
First we find the left limit of f(x) at x = 0
When x = 0–, |x| = – x
∴ f(x) = sin (– x) = – sin x
f(0) = – sin 0 = 0
`f"'"(0^-) = lim_(x ->0^-) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^-) (- sinx - 0)/x`
= `- lim_(x -> 0^-) sinx/x`
`f"'"(0^-)` = – 1 ........(1)
Next we find the right limit of f (x) at x = 0
When x = 0+ |x| = x
∴ f(x) = sin x
f(0) = sin 0 = 0
`f"'"(0^+) = lim_(x ->0^+) (f(x) - f(0))/(x - 0)`
= `lim_(x -> 0^+) (sinx - 0)/x`
= `lim_(x -> 0^+) sinx/x`
`f"'"(0^+)` = 1 ........(2)
From equations (1) and (2), we get
f’(0–) ≠ f'(0+)
∴ f(x) is not differentiable at x = 0.
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