see

limh0 f(x+h)-f(x)/h= limh0 [f(x) + f(h) - f(x)]/h   (as it is given that f(x+y) = f(x) + f(y)

 = limh0  f(h)/h          now f(h) = h^2g(h) (bcoz it's given that f(x) = x^2g(x))

=> limh0  f(h)/h  =  limh0 hg(h) =  0

hope now u have understood