Limits · Definition

Asymptotes

calculuslimitsgraphs

Before you start you need to know

Asymptotes are lines that guide the shape of a graph away from its problematic points.

Definition

An asymptote is a line the graph approaches without touching it. The line \(x=a\) is vertical if \(f\) tends to \(\pm\infty\) as \(x\to a\); the line \(y=L\) is horizontal if \(f(x)\to L\) as \(x\to\pm\infty\).

\[\begin{gathered}x=a:\ \lim_{x\to a}f(x)=\pm\infty\\ y=L:\ \lim_{x\to\pm\infty}f(x)=L\end{gathered}\]

An asymptote is a line the graph approaches without touching in the limit. Vertical ones appear where the function blows up (infinite limit at a point) and horizontal ones describe the behavior at \(\pm\infty\).

That is why they are computed with one-sided limits and limits at infinity: they are the geometric reading of the two previous concepts.

A function may cross its horizontal asymptote at finite points; what matters is what happens "very far away".

Think about it

Where is the vertical asymptote of \(\frac{1}{x-2}\)?

At x = 2, because the denominator vanishes there and the one-sided limits are infinite.

Exercises 10

  1. What are the asymptotes of \(f(x)=\frac{2 x + 1}{x - 3}\)?
  2. What are the asymptotes of \(f(x)=\frac{3 x}{x + 2}\)?
  3. What are the asymptotes of \(f(x)=\frac{x^{2} + 1}{x^{2} - 4}\)?
  4. What are the asymptotes of \(f(x)=\frac{1}{x+5}\)?
  5. What is a vertical asymptote of \(f\) at \(x=a\)?
  6. What are the asymptotes of \(f(x)=\frac{x-1}{x^{2}-1}\)?
  7. What are the asymptotes of \(f(x)=\frac{2}{(x-1)^{2}}\)?
  8. Which statement about \(f(x)=\frac{x}{x^{2}+1}\) is correct?
  9. What does \(\lim_{x\to\infty}f(x)=L\) mean?
  10. What are the asymptotes of \(f(x)=\frac{3x+6}{x^{2}-x-6}\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.