will d following function be differentiable at x = 0 ???

1. cos(|x|)+|x|

2. sin(|x|)-|x|

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- Aug 10th 2009, 02:29 PMgaurav1292check d differentiability
will d following function be differentiable at x = 0 ???

1. cos(|x|)+|x|

2. sin(|x|)-|x| - Aug 10th 2009, 03:08 PMTwig
Hi

First one:

If $\displaystyle h $ approaches zero from the right, we have:

$\displaystyle \frac{f(x+h)-f(x)}{h} = \frac{cos(h)+h-1}{h}=\frac{cos(h)-1}{h}+\frac{h}{h} \to 1 \mbox{ as } h \to 0 $

Now we approach from the left. Note that $\displaystyle cos(h)=cos(-h) $ , $\displaystyle cos(x)$ is an even function.

$\displaystyle \frac{cos(h)-h-1}{h}=\frac{cos(h)-1}{h}-\frac{h}{h} \to -1 \mbox{ as } h \to 0 $

So the function is not differentiable at $\displaystyle x = 0 $ .

The second one you basically do the same thing.

$\displaystyle \frac{sin(h)-h}{h} \to 1-1 = 0 $ and

$\displaystyle \frac{-sin(h)+h}{h} \to -1 +1 = 0 $ .

So the second one is differentiable.