Let-f-x-x-x-1-Find-lim-x-0-f-x-f-0-x-0-and-lim-x-0-f-x-f-0-x-0-Hence-determine-whether-f-x-is-differentiable-at-x-0-

Question Number 161463 by ZiYangLee last updated on 18/Dec/21

Let f (x) = x + ∣ x ∣ - 1 .

Find \lim_{x \to 0^{+}} \frac{f (x) - f (0)}{x - 0} and \lim_{x \to 0^{-}} \frac{f (x) - f (0)}{x - 0} .

Hence, determine whether f (x) is

differentiable at x = 0 .

Answered by TheSupreme last updated on 18/Dec/21

f (x) = {\begin{cases} - 1 i f x < 0 \\ 2 x - 1 i f x > 0 \end{cases}

f (0) = - 1

\frac{f (x) - f (0)}{x} = {\begin{cases} \frac{0}{x} x < 0 \\ 2 x > 0 \end{cases}

Answered by mathmax by abdo last updated on 18/Dec/21

\lim_{x \to o^{+}} \frac{f (x) - f (0)}{x - 0} = \lim_{x \to 0^{+}} \frac{2 x - 1 + 1}{x} = \lim_{x \to 0^{+}} \frac{2 x}{x} = 2

\lim_{x \to 0^{-}} \frac{f (x) - f (0)}{x - 0} = \lim_{x \to 0^{-}} \frac{x - x - 1 + 1}{x} = \lim_{x \to 0^{-}} \frac{0}{\bar{0}} = 0

f is not differenciale at x_{0} = 0