Question

Determine whether the following function is differentiable at the indicated values.

f(x) = sin |x| at x = 0

Answer 1

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.