will d following function be differentiable at x = 0 ???
1. cos(|x|)+|x|
2. sin(|x|)-|x|
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.