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Question: A zoologist studying social behavior models the interaction between two animals with the function $ I(u) = u - \frac{u^3}{9} $. If $ n $ is a positive integer, define $ x_n $ as the value of $ u $ such that $ I(x_n) = \frac{1}{n} $. Find $ \lim_{n \to \infty} x_n $.
Solution: We are given $ I(u) = u - \frac{u^3}{9} $, and $ I(x_n) = \frac{1}{n} $. So:
x_n - \frac{x_n^3}{9} = \frac{1}{n}.

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