Short on time? This page is the recap. For explanations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → ∞) f(x) means “the limit as x increases without bound of f(x)”.
Recap
- A limit at infinity asks what f(x) approaches as x grows without bound (x → ∞) or decreases without bound (x → −∞).
- Limits at infinity describe end behaviour: levelling off, growing without bound, or oscillating.
- y = L is a horizontal asymptote if f(x) → L as x → ∞ or as x → −∞. At most two: one per end.
- A graph may cross its horizontal asymptote. The asymptote only describes the far ends.
- To compare growth, find lim (x → ∞) f(x)/g(x): ∞ means f dominates, 0 means g dominates, a positive number means the same rate.
Key relationships
| Situation | Result as x → ±∞ |
|---|---|
| 1/xⁿ, n a positive integer | → 0 |
| Rational, degree top < degree bottom | → 0; asymptote y = 0 |
| Rational, equal degrees | → ratio of leading coefficients |
| Rational, degree top > degree bottom | no finite limit; no horizontal asymptote |
| √(x²) | the absolute value of x: equals x for x > 0, −x for x < 0 |
| eˣ | → ∞ as x → ∞; → 0 as x → −∞ |
| Growth order for large x | ln x < xᵖ (p > 0) < bˣ (b > 1) |
Assumptions behind the method
- Divide every term on top and bottom by the same nonzero quantity; this does not change the value.
- When x → −∞, the quantity you divide by inside a root must respect the sign: divide by −x, which is positive.
- The growth order is used here as a known fact. It is proved later with L’Hospital’s rule (Topic 4.7).
Mistakes to avoid
- Substituting ∞, or treating ∞/∞ as 1.
- Writing √(x²) = x when x → −∞. The sign of the answer flips.
- Using constant terms instead of leading terms for large x.
- Calling y = ∞ an asymptote. An infinite limit at an end means no horizontal asymptote there.
- Assuming both ends match. Roots, eˣ and arctan x can give different limits at the two ends.
- Judging growth from small x. x³ beats 2ˣ at x = 5, but 2ˣ wins in the long run.
Quick self-check
- Find lim (x → ∞) (3x − 7x³)/(2x³ + x). (−7/2)
- Find lim (x → −∞) √(x² + 5)/(2x). (−1/2: the root is positive and 2x is negative)
- Find lim (x → ∞) (ln x)/x. (0: x grows faster than ln x)
Next: practice questions.