Question

What is the limit of `(5x-9)/(4x^3+1)` as x goes to infinity?

Answer 1

0

Explanation:

It's clear from the graph shown below that as x gets bigger and bigger, the line gets closer and closer to 0. I know it's not very rigorous, but it works.

graph{(5x-9)/(4x^3+1) [-10, 10, -5, 5]}

Answer 2

See the explanation section.

Explanation:

For all `x` other than `0`, we have:

`(5x-9)/(4x^3+1) = (x^3(5/x^2-9/x^3))/(x^3(4+1/x^3))`

`= (5/x^2-9/x^3)/(4+1/x^3)`

`lim_(xrarroo) (5/x^2-9/x^3) = 0`

and
`lim_(xrarroo) (4+1/x^3) = 4`.

So,

`lim_(xrarroo) (5x-9)/(4x^3+1) = lim_(xrarroo) (5/x^2-9/x^3)/(4+1/x^3)`

`= 0/4 = 0`