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

1 Answer

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](useruploads.socratic.orguseruploads.socratic.org)

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