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Thank you @Sasha. The thought process behind my question is something like this. Suppose we have a family of stable, smooth genus 0 curves $(C, p, q, r_\lambda)$ with $r_\lambda \to q$ as $\lambda \to 0.$ Then we have a process for replacing the unstable fiber above 0 with a stable curve, and we have some good idea for why and what situations we do this in. So there may be circumstances/reasons why we don't use the "obvious" thing as the limit, and I am interested in peoples' thoughts on this.