Differentiation and continuity

[macjack]

Can anyone tell me whether sin|x| and cos|x| is differentiable at x=0 ?
As far as i know, cos(x) and sin(x) is differentiable at all x.

If i try to solve this,
lim h->0 (f(x+h) - f(x))/h
when x=0, and substitute cos|x| for f(x).

lim h->0 (cos|h| - cos|0|)/(h) = 1,

so cos|x| is differentiable at x=0 right ?

And for sin|x| ...if i continue doing the same as my previous try,
lim h->0 (sin|h| - sin|0|)/(h) = limh->0 sin|h|/h = 1 .

So both are differenatiable at x=0 as per my explanation,
but the answer given is in another way.

Can you please let me whether it is correct or not ?

Thanks

Mac

 

[NateTG]

Your answer is (at least partially) incorrect.

For the derivative to exist, the limit must exist from both sides, and be the same. Also, your limit on the cosine of the absolute value is incorrect.