Convergent Real Sequence/Examples/2 n^3 - 3 n over 5 n^3 + 4 n^2 - 2

From ProofWiki
Jump to navigation Jump to search

Example of Convergent Real Sequence

$\displaystyle \lim_{n \mathop \to \infty} \paren {\dfrac {2 n^3 - 3 n} {5 n^3 + 4 n^2 - 2} } = \dfrac 2 5$


Proof

\(\displaystyle \dfrac {2 n^3 - 3 n} {5 n^3 + 4 n^2 - 2}\) \(=\) \(\displaystyle \dfrac {2 - \dfrac 3 {n^2} } {5 + \dfrac 4 n - \dfrac 2 {n^3} }\) dividing top and bottom by $n^3$
\(\displaystyle \) \(\to\) \(\displaystyle \dfrac {2 - 0} {5 + 0 - 0}\) \(\displaystyle \text {as $n \to \infty$}\) Sequence of Powers of Reciprocals is Null Sequence
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 2 5\)

$\blacksquare$


Sources