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Let \( \displaystyle f \) and \( \displaystyle g \) be analytic on a region \( \displaystyle A \), both having zeros of order \( \displaystyle k \) at \( \displaystyle z_{0} \in A \). Show that \( \displaystyle \frac{f}{g} \) has a removable singularity at \( \displaystyle z_{0} \) and that \( \displaystyle \lim_{z \to z_{0}} \frac{f(z)}{g(z)} = \frac{ f^{(k)}(z_{0}) }{ g^{(k)}(z_{0}) } \).