Short on time? This page is the recap. For explanations, graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → a) f(x) means “the limit as x approaches a of f(x)”. x → a⁻ is from the left; x → a⁺ is from the right.
Recap
- An infinite limit describes values that grow without bound near x = a: “= ∞” if positive, “= −∞” if negative.
- ∞ is not a number, so such a limit does not exist; the symbol says how it fails.
- Write a two-sided “lim (x → a) f(x) = ∞” only when both sides go to +∞ (or both to −∞). Otherwise give the one-sided limits.
- Vertical asymptote: x = a is one if at least one one-sided limit at a is ±∞.
- Justify it in words: “Because lim (x → a⁺) f(x) = ∞, x = a is a vertical asymptote.”
Key relationships
| Situation at x = a | What it tells you | Next step |
|---|---|---|
| Substitution gives a number, bottom ≠ 0 | Ordinary limit | Done |
| 0/0 | Not decided yet | Factor and cancel, then substitute again |
| 0/0, factor cancels, bottom now ≠ 0 | Hole, finite limit | Give the height of the hole |
| Nonzero/0 (before or after cancelling) | Vertical asymptote | Sign check on each side |
| (x − a) to an odd power in the bottom | Sign flips across a | One-sided limits are opposite |
| (x − a) to an even power in the bottom | Same sign on both sides | Two-sided ±∞ |
Standard infinite limits: lim (x → 0⁺) ln x = −∞; lim (x → π/2⁻) tan x = ∞ and lim (x → π/2⁺) tan x = −∞; lim (x → 0) 1/x² = ∞.
Assumptions behind the method
- The sign check needs the top to tend to a nonzero number. If the top also tends to 0, simplify first.
- “Slightly left” and “slightly right” mean values close enough to a that no other zero of the top or bottom lies in between.
- Being undefined at a is not enough for an asymptote. You need an infinite limit (for example, x/ln x → 0 as x → 0⁺, so x = 0 is not an asymptote).
Mistakes to avoid
- Calling every zero of the denominator an asymptote. Cancel common factors first: some are holes.
- “Nonzero/0 = 0”. A fixed number over something tiny is huge.
- Guessing +∞. Check the signs of the top and the bottom on each side.
- Writing “lim = ∞” when the sides disagree. Use one-sided limits.
- Thinking both sides must be infinite. One side is enough (ln x at 0).
Quick self-check
- Find lim (x → 2⁻) (x + 4)/(x − 2) and lim (x → 2⁺) (x + 4)/(x − 2). (−∞ and +∞: the top tends to 6 and x − 2 changes sign at 2)
- Where does (x² − 4)/(x² + x − 6) have a hole, and where a vertical asymptote? (It simplifies to (x + 2)/(x + 3) for x ≠ 2. Hole at (2, 4/5); vertical asymptote at x = −3)
- Find lim (x → −5) −1/(x + 5)². (−∞: the bottom is positive on both sides and the top is −1)
Next: practice questions.