📌 Topic 1.9 — Vertical Asymptotes
Holes vs. Vertical Asymptotes — The Core Rule
Both occur when the denominator = 0 · which one depends on whether the factor cancels
Holes vs. Vertical Asymptotes — Key Distinction
🕳️ Hole (Removable Discontinuity)
A hole occurs when a factor in the denominator cancels out with a matching factor in the numerator.
The function is undefined at that x-value, but the limit exists (from both sides it approaches the same finite value).
f(x) = (x−1)(x+2)x−1
g(x) = (x−1)³2(x−1)²
Both have a hole at x = 1
📏 Vertical Asymptote
A vertical asymptote occurs when a factor in the denominator does not cancel with any factor in the numerator.
Near this x-value, the function heads to +∞ or −∞ from one or both sides.
f(x) = (x−3)(x+2)x−1
g(x) = (x−1)(x+2)(x−1)²
Both have a VA at x = 1
The Test: Cancel or Not Cancel?
Step 1: Fully factor both numerator and denominator.
Step 2: For each value where the denominator = 0, check if that factor also appears in the numerator.
→ Factor cancels → Hole
→ Factor does NOT cancel → Vertical Asymptote
Step 3: Simplify (cancel common factors) and work with the simplified form.
Identifying Holes and VAs — Example 1
Factor completely, then check each denominator factor
f(x) = (x−2)(x+3)(x+3)(x−5)
🕳️ Hole at x = −3
(x+3) cancels with numerator
(x+3) cancels with numerator
📏 VA at x = 5
(x−5) does not cancel
(x−5) does not cancel
y = (x+1)(x−2)²(x−2)(x+1)²
🕳️ Hole at x = 2
(x−2) cancels: num has ², den has ¹ → one (x−2) remains in num
(x−2) cancels: num has ², den has ¹ → one (x−2) remains in num
📏 VA at x = −1
(x+1) has one left over in denominator after canceling
(x+1) has one left over in denominator after canceling
g(x) = 1x³ + 4x = 1x(x² + 4)
📏 VA at x = 0
Factor x doesn't cancel (numerator = 1)
Factor x doesn't cancel (numerator = 1)
x²+4 is always > 0 — never equals 0, so no additional VA or hole from that factor.
📌 Topic 1.10 — Limit Behavior Near Holes & VAs
One-Sided Limits — What They Tell You
Equal finite limits → hole · limits go to ±∞ → vertical asymptote
Limit Behavior at Holes vs. Vertical Asymptotes
🕳️ Hole — Limits Are Equal & Finite
If lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = L (a finite number), then there is a hole at x = a.
The function is undefined at x = a, but approaches the same value from both sides. The "hole" is at the point (a, L).
Example
lim(x→2⁻) = −6
and
lim(x→2⁺) = −6
→ Hole at (2, −6)
📏 VA — Limits Go to ±∞
If one or both one-sided limits go to +∞ or −∞ at x = a, then there is a vertical asymptote at x = a.
The two sides can go in the same direction (both +∞ or both −∞) or opposite directions — depends on multiplicity.
Example
lim(x→2⁻) = +∞
and
lim(x→2⁺) = −∞
→ VA at x = 2 (odd multiplicity)
VA Direction — Odd vs. Even Multiplicity
Odd multiplicity (e.g. (x−a)¹): the function goes to +∞ on one side and −∞ on the other — it crosses through the asymptote direction.
Even multiplicity (e.g. (x−a)²): the function goes to +∞ on both sides (or −∞ on both sides) — it bounces off the asymptote in the same direction.
To determine which direction: substitute a value just to the left and right of the VA into the simplified function and check the sign.
Finding One-Sided Limits — Example 2
Determine limits from both sides of the critical value
📌 Example 2a — Find lim(x→2⁻) and lim(x→2⁺) for g(x) = (x−2)(x+4)(x−2)(x−3)
1
Factor check: (x−2) appears in both numerator and denominator → it cancels → hole at x = 2 (not a VA).2
Simplify: g(x) = x+4x−3 for x ≠ 2.3
Evaluate limit: plug x = 2 into the simplified form: 2+42−3 = 6−1 = −6.4
Both one-sided limits equal the same finite value: lim(x→2⁻) = lim(x→2⁺) = −6.Hole at (2, −6) · lim(x→2⁻) = lim(x→2⁺) = −6
📌 Example 2b — Find lim(x→2⁻) and lim(x→2⁺) for h(x) = (x−4)(x−2)(x−2)²(x−1)
1
Factor check: (x−2) appears once in numerator and twice in denominator. Cancel one → one (x−2) remains in denominator → VA at x = 2 (does not fully cancel).2
Simplified: h(x) = x−4(x−2)(x−1) for x ≠ 2.3
Sign analysis near x = 2: numerator at x≈2: (2−4) = −2 → negative. (x−1) at x≈2: (2−1) = 1 → positive.4
Left side (x→2⁻): (x−2) is small negative → denominator = (−)(+) = negative. So h ≈ −2(−)(+) → +∞.5
Right side (x→2⁺): (x−2) is small positive → denominator = (+)(+) = positive. So h ≈ −2(+)(+) → −∞.VA at x = 2 · lim(x→2⁻) = +∞ · lim(x→2⁺) = −∞
Building an Equation from Limit Properties — Example 3
Work backwards from limit behavior to construct a rational function
📌 Example 3a — A rational function has lim(x→3⁻) = lim(x→3⁺) = 5 and lim(x→1⁻) = −∞, lim(x→1⁺) = +∞. Find a possible equation.
1
Equal finite limits at x = 3 → hole at x = 3 → factor (x−3) must cancel → include (x−3) in both numerator and denominator.2
Opposite-direction limits at x = 1 (−∞ left, +∞ right) → VA at x = 1 with odd multiplicity → include (x−1) in denominator only.3
Simplest form: f(x) = k(x−3)(x−3)(x−1). After canceling: f(x) = kx−1.4
Use the hole value to find k: plug x = 3 into simplified → k3−1 = k2 = 5 → k = 10.f(x) = 10(x−3)(x−3)(x−1)
Reading Limit Statements — Quick Guide
lim(x→a⁻) = lim(x→a⁺) = L (finite) → Hole at (a, L). Factor (x−a) cancels completely.
lim(x→a⁻) = +∞ and lim(x→a⁺) = −∞ (or vice versa) → VA, odd multiplicity. Signs flip across x = a.
lim(x→a⁻) = lim(x→a⁺) = +∞ (or both −∞) → VA, even multiplicity. Signs stay the same across x = a.
Quick Reference
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🔑 Key Rules
Factor cancels fully → Hole
Factor does NOT cancel → VA
Equal finite limits → Hole at (a, L)
Limits → ±∞ → Vertical Asymptote
Odd mult VA: limits go in opposite directions
Even mult VA: limits go in same direction
⚠️ Common Mistakes
❌ Calling a hole a VA (or vice versa)
Always check: does the factor fully cancel? If yes → hole. If any power remains in denominator → VA. Count powers carefully.
❌ Forgetting to simplify before evaluating the hole
To find the y-coordinate of a hole, plug the x-value into the simplified function — not the original. The original is undefined there.
❌ Assuming equal limits always means hole
Equal limits → hole only if the limit is a finite number. If both limits go to +∞ (even mult VA), that's still a VA — the limits are equal but not finite.
❌ Wrong VA direction — not checking sign
Don't guess the direction. Substitute a test value just left and right of the VA into the simplified expression to determine the sign on each side.