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 December 27th, 2014, 07:37 AM #1 Senior Member   Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 Differentiation Show that $\displaystyle y=\left |x-2 \right |$ is not differentiable at $\displaystyle x=2$ How to show? December 27th, 2014, 08:33 AM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 3,031 Thanks: 1620 derivative at a point $\displaystyle x = a$ is defined as $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$ for $\displaystyle f(x) = |x-2|$, show that the limit ... $\displaystyle \lim_{x \to 2} \frac{f(x)-f(2)}{x-2}$ ... does not exist. December 27th, 2014, 01:34 PM   #3
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 Originally Posted by skeeter derivative at a point $\displaystyle x = a$ is defined as $\displaystyle f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a}$ for $\displaystyle f(x) = |x-2|$, show that the limit ... $\displaystyle \lim_{x \to 2} \frac{f(x)-f(2)}{x-2}$ ... does not exist.
The limits for x > 2 and x < 2 are different. December 27th, 2014, 06:01 PM #4 Senior Member   Joined: Oct 2014 From: EU Posts: 224 Thanks: 26 Math Focus: Calculus You can consider the function as a piecewise function: $\displaystyle f(x) = \left \{ \begin{matrix} f_a(x) = x - 2 &\text{if}\,x \ge 2 \\ f_b(x) = -x+2 &\text{if}\,x < 2 \end{matrix}\right.$ Now, you can see that $\displaystyle f(x)$ is continous at $\displaystyle x=2$ evaluating the limits for $\displaystyle f_a(x)$ and $\displaystyle f_b(x)$ when $\displaystyle x$ tends to 2, but each derivative is different: 1 and -1, so $\displaystyle f(x)$ is not differentiable when $\displaystyle x = 2$. Last edited by szz; December 27th, 2014 at 06:21 PM. Tags calculus, differentiation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Happy Math Books 0 December 13th, 2014 03:20 AM KO1337 Academic Guidance 2 May 22nd, 2014 04:41 AM pingpong Math Books 0 August 13th, 2013 03:25 PM tdod Calculus 3 December 14th, 2011 11:24 AM kingkos Algebra 0 December 31st, 1969 04:00 PM

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