8 The sequence \(\left\{ u _ { n } \right\}\) of positive real numbers is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = \frac { 2 u _ { n } + 3 } { u _ { n } + 2 }\) for \(n \geqslant 1\).
- Prove by induction that \(u _ { n } ^ { 2 } - 3 < 0\) for all positive integers \(n\).
- By considering \(u _ { n + 1 } - u _ { n }\), use the result of part (a) to show that \(u _ { n + 1 } > u _ { n }\) for all positive integers \(n\).
The sequence \(\left\{ u _ { n } \right\}\) has a limit for \(n \rightarrow \infty\).
- Find the limit of the sequence \(\left\{ u _ { n } \right\}\) as \(n \rightarrow \infty\).
- Describe as fully as possible the behaviour of the sequence \(\left\{ u _ { n } \right\}\).
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