Asymptotes

Asymptotes are straight lines that the graph of a function approaches indefinitely: they are all identified through limits.

In brief — The three types of asymptote

• Horizontal asymptote: $y=L$ if $\displaystyle\lim_{x\to\pm\infty}f(x)=L$.
• Vertical asymptote: $x=x_0$ if $\displaystyle\lim_{x\to x_0^\pm}f(x)=\pm\infty$.
• Oblique asymptote: $y=mx+q$ with $\displaystyle m=\lim_{x\to\pm\infty}\frac{f(x)}{x}$ and $\displaystyle q=\lim_{x\to\pm\infty}\bigl(f(x)-mx\bigr)$.

The function $f(x) = x + \dfrac{1}{x}$ illustrates two asymptotes together: a vertical asymptote at $x=0$ and an oblique asymptote $y=x$ as $x\to\pm\infty$.

The function $f(x) = x + 1/x$ has oblique asymptote $y=x$ (as $x\to\pm\infty$) and vertical asymptote $x=0$.

Links

Topics: Limits
Concepts: Asymptote · Oblique asymptote · Horizontal asymptote · Vertical asymptote · Limit
Skills: Interpreting graphs · Studying functions