Asymptote/Examples/x^3 over x^2 + x + 1

Examples of Asymptotes

Asymptote ($\color { red } {\text {red} }$) to $\dfrac {x^3} {x^2 + x + 1}$ ($\color { blue } {\text {blue} }$)

Consider the curve whose equation in the Cartesian plane is given by:

$y = \dfrac {x^3} {x^2 + x + 1}$

This has $1$ slant asymptote:

the straight line $y = x - 1$.

Proof

The original equation can be rewritten:

$y = x - 1 + \dfrac 1 {x^2 + x + 1}$

from which the result follows.

$\blacksquare$

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): asymptote
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): asymptote