Here are some examples to show that the case of holomorphic functions really is a special case.

1. Notice that if we let $f_n(z)=\frac{z}{n}$, then this function only has one zero, but the limit is equal to $0$.
2. Consider the real numbers, and a function of the kind $f_n(x)=x^2+\frac{1}{n}$. these functions have no zeros, because this is an irreducible polynomial. But it's limit is $x^2$ which has two zeros.