Convergent Real Sequence/Examples/n^3 + 5 n^2 + 2 over 2 n^3 + 9

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Example of Convergent Real Sequence

$\displaystyle \lim_{n \mathop \to \infty} \paren {\dfrac {n^3 + 5 n^2 + 2} {2 n^3 + 9} } = \dfrac 1 2$

Proof

 $\displaystyle \dfrac {n^3 + 5 n^2 + 2} {2 n^3 + 9}$ $=$ $\displaystyle \dfrac {1 - \dfrac 5 n + \dfrac 2 {n^3} } {2 + \dfrac 9 {n^3} }$ dividing top and bottom by $n^3$ $\displaystyle$ $\to$ $\displaystyle \dfrac {1 + 0 + 0} {2 + 0}$ $\displaystyle \text {as n \to \infty}$ Sequence of Powers of Reciprocals is Null Sequence $\displaystyle$ $=$ $\displaystyle \dfrac 1 2$

$\blacksquare$