Limits - Calculus?

If the limit of a product is the product of the limits (lim x-->n (f(x)*g(x))=limx-->n f(x)*limx-->n g(x) ) [i] and the limit of a constant is zero (limx-->n C=0) [ii] then why it is wrong to say that, for example, limx-->0 5sinx/3x = limx-->0 (sinx/x)*(5/3) = limx-->0 (sinx/x) * limx-->0 5/3 = 0?

So [1] is true only for f(x) and g(x) not constant?

Thanks