📌 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)³ / [2(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)
———————————
(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
———————————
(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) = 1/(x³ + 4x)
= 1/[x(x² + 4)]
= 1/[x(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 — Holes and Limit Notation
Limit Behavior Near VAs and Holes
VAs → ±∞ (one or both sides) · Holes → finite value from both sides
Limit Behavior: Vertical Asymptote vs. Hole
📏 Near a Vertical Asymptote
Left and/or right limits go to +∞ or −∞. Determine the sign using a sign chart near the VA.
Left and right limits may have opposite signs (one +∞, one −∞), or both the same.
lim(x→1⁻) f(x) = −∞
lim(x→1⁺) f(x) = +∞
The two limits are different → VA confirmed
🕳️ Near a Hole
Both left and right limits equal the same finite value. Plug the x-value into the simplified (cancelled) form.
The function is undefined there, but approaches the same finite y-value from both sides.
lim(x→2⁻) g(x) = −6
lim(x→2⁺) g(x) = −6
Both limits equal −6 → Hole confirmed
How to find the sign near a VA
Pick an x-value just to the left of the VA (like VA − 0.001) and determine if the expression is positive or negative. Do the same for the right side. This gives you the sign of the limit (+∞ or −∞).
Shortcut: substitute the VA value and track the sign of each factor. Use ( ) to denote positive and (−) to denote negative.
Writing Left and Right Limits — Example 2
As x approaches 2 for each function — VA or hole? What are the limits?
📌 Example 2a — f(x) = (x−1)(x+3)/(x−2) · Write left and right limits as x → 2.
(x−1)(x+3) / (x−2)
1
x=2 makes denominator = 0. Numerator at x=2: (1)(5) = 5 ≠ 0. Factor (x−2) does NOT cancel → Vertical Asymptote at x=2.2
Sign analysis near x=2:
Sign of (x−1)(x+3)/(x−2) near x = 2
x → 2⁻:
(+)(+)/(−) → negative → −∞
x → 2⁺:
(+)(+)/(+) → positive → +∞
Near x=2: (x−1) ≈ (+1) · (x+3) ≈ (+5) · (x−2) → 0⁻ or 0⁺
lim(x→2⁻) f(x) = −∞ · lim(x→2⁺) f(x) = +∞
📌 Example 2b — g(x) = (x−2)(x+4)/[(x−2)(x−3)] · Write left and right limits as x → 2.
(x−2)(x+4) / [(x−2)(x−3)]
1
Factor (x−2) cancels from both numerator and denominator → Hole at x=2. Simplified: g(x) = (x+4)/(x−3).2
Evaluate simplified form at x=2: (2+4)/(2−3) = 6/(−1) = −6.3
Both sides approach the same finite value → left limit = right limit = −6.lim(x→2⁻) g(x) = −6 · lim(x→2⁺) g(x) = −6 (Hole at x=2, y=−6)
📌 Example 2c — h(x) = (x−4)(x−2)/[(x−2)²(x−1)] · Write left and right limits as x → 2.
(x−4)(x−2) / [(x−2)²(x−1)]
1
Factor (x−2) appears once in numerator and twice in denominator. Cancel one (x−2) each side → simplified: h(x) = (x−4)/[(x−2)(x−1)]. One (x−2) remains in denominator → Vertical Asymptote at x=2.2
Sign analysis near x=2 using simplified form (x−4)/[(x−2)(x−1)]:
Sign near x = 2
x → 2⁻:
(x−4)≈−2 · (x−2)→0⁻ · (x−1)≈1 → (−)/[(−)(+)] → (−)/(−) = positive → +∞
x → 2⁺:
(x−4)≈−2 · (x−2)→0⁺ · (x−1)≈1 → (−)/[(+)(+)] = negative → −∞
lim(x→2⁻) h(x) = +∞ · lim(x→2⁺) h(x) = −∞
Building Equations from Limit Properties — Example 3
Work backward: equal limits → hole · unequal limits → VA
Strategy for Building a Rational Function from Limit Properties
Equal limits at x=a (lim from left = lim from right = finite number) → hole at x=a → factor (x−a) cancels from numerator and denominator.
Unequal limits at x=a (one +∞, one −∞, or both ±∞) → VA at x=a → factor (x−a) stays in denominator.
Use a constant k in the numerator to match the given limit value, then solve for k.
Example 3a
lim(x→3⁻) = 5 · lim(x→3⁺) = 5 → equal → Hole at x=3
lim(x→1⁻) = −∞ · lim(x→1⁺) = +∞ → VA at x=1
lim(x→1⁻) = −∞ · lim(x→1⁺) = +∞ → VA at x=1
Build: use (x−3) in both numerator and denominator (hole), and (x−1) only in denominator (VA):
f(x) = k(x−3) / [(x−3)(x−1)]
Cancel (x−3): simplified f(x) = k/(x−1). At x=3: k/(3−1) = k/2 = 5 → k = 10
f(x) = 10(x−3) / [(x−3)(x−1)]
Example 3b
lim(x→−2⁻) = 4 · lim(x→−2⁺) = 4 → equal → Hole at x=−2
lim(x→−1⁻) = +∞ · lim(x→−1⁺) = +∞ → same direction → VA at x=−1, even multiplicity
lim(x→−1⁻) = +∞ · lim(x→−1⁺) = +∞ → same direction → VA at x=−1, even multiplicity
Both sides +∞ near x=−1 → even-mult VA. Use (x+1)² in denominator. Hole at x=−2 → (x+2) cancels:
f(x) = k(x+2) / [(x+2)(x+1)²]
Cancel (x+2): simplified f(x) = k/(x+1)². At x=−2: k/(−2+1)² = k/1 = 4 → k = 4
f(x) = 4(x+2) / [(x+2)(x+1)²]
Quick Reference
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🕳️ Holes
Factor cancels → hole at that x-value
lim from left = lim from right = finite y-value
Plug x into the SIMPLIFIED (cancelled) form
Graph has a small open circle at that point
📏 Vertical Asymptotes
Factor does NOT cancel → VA at that x-value
Limits → ±∞ · track sign near the VA
Even mult VA: both sides same direction (±∞)
Odd mult VA: sides go opposite directions
⚠️ Common Mistakes
❌ Hole AND VA at the same x-value
You can't have both at the same point. If the factor cancels → hole. If it doesn't → VA. Never both.
❌ Not fully factoring before deciding
Always factor COMPLETELY first. A factor like (x²−4) should become (x−2)(x+2) — they may cancel separately.
❌ Plugging into original (unfactored) form for hole value
For the y-value of a hole, plug into the SIMPLIFIED form (after canceling). The original form is undefined there.
❌ Assuming even-mult VA means no asymptote
Even multiplicity VA still means the function goes to ±∞ — both sides go the SAME direction (both +∞ or both −∞).