Question

Lim
n->infinity 4n^3-5/9n^3+7

Answer 1

`\lim_(n\rightarrow +\infty ) (4n^(3)-5)/(9n^(3)+7)=(4)/(9)`

Explanation:

`\lim_(n\rightarrow +\infty ) (4n^(3)-5)/(9n^(3)+7)`

Pull out n³

`=\lim_(n\rightarrow +\infty ) (n^(3)* \left( 4-(5)/(n^3) \right) )/(n^(3)* \left( 9+(7)/(n^3) \right) )`

Simplify by n³

`=\lim_(n\rightarrow +\infty ) ( \left( 4-(5)/(n^3) \right) )/( \left( 9+(7)/(n^3) \right) )`

Remark :
`\lim_(n\rightarrow +\infty ) (5)/(n^3) \right) } = 0 \ \. \text{and} \ \ \lim_(n\rightarrow +\infty ) (7)/(n^3) \right) } = 0`

Then

`\lim_(n\rightarrow +\infty ) ( \left( 4-(5)/(n^3) \right) )/( \left( 9+(7)/(n^3) \right) )=(4-0)/(9+0)`

`=(4)/(9)`