Analytic l’Hôpital rule from a derivative-ratio limit

Let \(D\subset\mathbb C\) be a domain and let \(f,g:D\to\mathbb C\) be analytic. Suppose \(z_0\in D\), \(f(z_0)=g(z_0)=0\), and \[ \lim_{z\to z_0}\frac{f'(z)}{g'(z)}\in\mathbb C \] exists. Show that \[ \lim_{z\to z_0}\frac{f(z)}{g(z)} \] exists and calculate it.