Question

If f(x) = 2sin|x|+ a(e^3|x|) is differentiable at x = 0, then value of a is ??

Answer 1

you know that

f'(x) = lim(h→0) (f(x+h)-f(x))/h
for f(x) to be differentiable, that limit must be the same both as
h→0- and as h→0+

so check the limit and decide. Post your work if you get stuck.
I'd pay special attention to x/|x|