Derivative and function ratios at equal-order zeros

Let \(D\subset\mathbb C\) be a domain containing \(0\) and let \(f,g\) be analytic on \(D\). Suppose both have zeros of order \(7\) at \(0\). Prove that \[ \lim_{z\to0}\frac{f'(z)}{g'(z)} \quad\text{and}\quad \lim_{z\to0}\frac{f(z)}{g(z)} \] both exist and are equal.