Limits · Definition

Limits at infinity

calculuslimits

Before you start you need to know

So far you have studied limits as \(x\) approaches a number. Here we look at what happens to \(f(x)\) as \(x\) grows without bound.

Definition

The limit of \(f\) as \(x\) tends to infinity describes what happens to \(f(x)\) when \(x\) grows without bound. For a quotient of polynomials, the result is decided by the quotient of the highest-degree terms.

\[\lim_{x\to\infty}\frac{a_n x^{n}+\cdots}{b_n x^{n}+\cdots}=\frac{a_n}{b_n}\]

Two different ideas appear with the same symbol. A limit at infinity (\(x\to\infty\)) asks what happens "very far away". An infinite limit (\(f(x)\to\infty\)) says the values grow without bound near a point.

For quotients of polynomials, the degree decides: the highest-degree term wins. That is why \(\frac{3x^2+1}{5x^2-x}\) tends to \(3/5\) as \(x\to\infty\).

Writing \(\lim f(x)=\infty\) does not mean the limit "exists" as a number; it is a way of describing how it diverges.

Think about it

What does \(\frac{2x+1}{x^2}\) tend to as \(x\to\infty\)?

To 0: the denominator has higher degree than the numerator.

Exercises 10

Compute the following limits

  1. \(\displaystyle \lim_{x\to\infty}\,\frac{3 x^{2} - 2 x + 1}{5 x^{2} + 4}\)
  2. \(\displaystyle \lim_{x\to\infty}\,\frac{2 x + 1}{x^{2} + 3}\)
  3. \(\displaystyle \lim_{x\to\infty}\,\frac{4 x^{3} - x}{x^{2} + 1}\)
  4. \(\displaystyle \lim_{x\to\infty}\,\frac{5 x - 3}{2 x + 7}\)
  5. \(\displaystyle \lim_{x\to\infty}\,\frac{\sqrt{x^{2} + 1}}{x}\)
  6. \(\displaystyle \lim_{x\to\infty}\,\frac{x^{2} + x}{3 x^{2} - 1}\)
  7. \(\displaystyle \lim_{x\to\infty}\,\frac{x^{2} - 3 x}{2 x + 1}\)
  8. \(\displaystyle \lim_{x\to\infty}\,\left(\sqrt{4 x^{2} + x} - 2 x\right)\)
  9. \(\displaystyle \lim_{x\to\infty}\,\frac{1}{x}\)
  10. \(\displaystyle \lim_{x\to\infty}\,\frac{2 x^{2} + x}{x^{3} - 1}\)

What comes next

References

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