Question

lim x approaches to 0 $\frac{(e^{ax} - 1) - (e^{bx} - 1 )}{x}$

Answer 1

if lt applies we get 0/0 but this is not exist 
so, apply LHospital rule
 Lt x -> 0   (a x e^ ax ) - (b x e^ bx)
                 ___________________________
                                1
now apply limit
you get,
          answer = a-b  (any power 0 is 1)