Question

How do you find the slant asymptote of `y=(x^3 - 4x^2 + 2x -5)/ (x^2 + 2)`?

Answer 1

`color(blue)("Slant Asymptote")`
`color(blue)(y=x-4)`

Explanation:

To find the slant asymptote, we divide `x^3-4x^2+2x-5` by `x^2+2`

The resulting quotient not including the remainder part represents the slant asymptote

Let us divide

`" " " " " " " " " " " "underline(x-4" " " " " " " " " " " " " ")`
`x^2+0*x+2|~x^3-4x^2+2x-5`
`" " " " " " " " " " " "underline(x^3+0x^2+2x" " " " " " " ")`
`" " " " " " " " " " " " " "-4x^2+0-5`
`" " " " " " " " " " " " " "underline(-4x^2+0x-8" " " " " ")`
`" " " " " " " " " " " " " " " " " " " " "" " " " +3`

Observe the quotient `x-4` so that our slant asymptote is

`y=x-4`

Kindly see the graph of `y=(x^3-4x^2+2x-5)/(x^2+2)" "`(colored red) and the slant asymptote `y=x-4" "`(colored blue).

![Desmos.com]()

God bless....I hope the explanation is useful.