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📚 Topic 1.7 · Unit 1

Rational Functions &
End Behavior

Horizontal asymptotes, slant asymptotes, and limit statements — determined by comparing degrees.

📋 Notes 🃏 Flashcards 🧠 Practice Quiz
📖
What is a Rational Function?
The definition and basic form
f(x) = p(x) / q(x)
A rational function is the quotient of two polynomials, where q(x) ≠ 0.
n = degree of p(x)
n is the degree of the numerator. The leading term is axⁿ.
d = degree of q(x)
d is the degree of the denominator. The leading term is bx^d.
🔑

End behavior is determined by leading terms only

As x → ±∞, lower-degree terms become negligible. Only the leading terms of the numerator and denominator matter:

f(x) ≈ axⁿbxd as x → ±∞

The relationship between n and d tells you everything about end behavior.

⚖️
The Three Cases
Compare n (numerator degree) vs d (denominator degree)
Case 1
n = d
Same degree → Horizontal Asymptote at y = ab (ratio of leading coefficients)
3x²5x² → y = 35
Case 2
n < d
Denominator dominates → Horizontal Asymptote at y = 0
2xx² → y = 0
Case 3
n > d
Numerator dominates → No horizontal asymptote. End behavior like y = abx^(n−d)
x³x² → behaves like y = x
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Special Case: Slant (Oblique) Asymptote

When n = d + 1 (numerator degree is exactly 1 more than denominator), the function has a slant asymptote.

The slant asymptote is parallel to the line y = abx where a and b are the leading coefficients.

⚠️ If n = d + 2 or more, there is no slant asymptote — the numerator dominates too much.

✏️
Example 1 — Identify Asymptote Type
Horizontal, slant, or neither?
f(x) = 3x²+4x−75x²−3
n=2, d=2 → n=d → Case 1
Leading coefficients: 3 and 5
HA: y = 35
y = 2x−5x²+3x+2
n=1, d=2 → n<d → Case 2
Denominator dominates
HA: y = 0
f(x) = 2x²−45x+9
n=2, d=1 → n=d+1
Slant! Leading ratio: 25
Slant asymptote (parallel to y = 25x)
y = 4x+58x−1
n=1, d=1 → n=d → Case 1
Leading coefficients: 4 and 8
HA: y = 48 = 12
f(x) = 3x²+3x−7
n=0, d=2 → n<d → Case 2
Denominator dominates
HA: y = 0
f(x) = −42x+1
n=0, d=1 → n<d → Case 2
Denominator dominates
HA: y = 0
↔️
Writing Limit Statements
Example 2 — describing end behavior formally
🎯

How to write limit statements

Identify the case, then determine what value (or ±∞) the function approaches:

• Case 1 (n=d): both limits = ab
• Case 2 (n<d): both limits = 0
• Case 3 (n>d): use end behavior of abx(n−d) — check sign carefully for x→−∞

📌 Example 2a — f(x) = 2x³+4x−16x³−x²+4
1
n=3, d=3 → n=d → Case 1. Leading coefficients: 2 and 6.
2
Ratio = 26 = 13. Horizontal asymptote y = 13.
3
Both limits equal the HA value.
lim x→−∞ f(x) = 13    lim x→+∞ f(x) = 13
📌 Example 2b — g(x) = 5x²−8x+92x³+x−1
1
n=2, d=3 → n<d → Case 2. Denominator dominates.
lim x→−∞ g(x) = 0    lim x→+∞ g(x) = 0
📌 Example 2c — h(x) = −3x⁴−x²+xx³+4x+4
1
n=4, d=3 → n>d → Case 3. End behavior like −31x(4−3) = −3x.
2
As x→−∞: −3x → −3(−∞) = +∞
3
As x→+∞: −3x → −3(+∞) = −∞
lim x→−∞ h(x) = +∞    lim x→+∞ h(x) = −∞
CaseConditionlim x→−∞lim x→+∞
1n = dabab
2n < d00
3 (odd n−d, ab > 0)n > d−∞+∞
3 (odd n−d, ab < 0)n > d+∞−∞
3 (even n−d, ab > 0)n > d+∞+∞
3 (even n−d, ab < 0)n > d−∞−∞
📐
Slant Asymptotes — Example 3
n = d + 1 exactly
⚠️

The Exact Requirement

A slant asymptote exists only when n = d + 1 — the numerator degree is exactly 1 more.

The slant asymptote is parallel to the line y = abx, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

If n = d + 2 or more → no slant, no horizontal asymptote (numerator dominates too much).

📌 Example 3 — Which function has a slant asymptote parallel to y = 12x?
I
f(x) = x²+32x²+x+6: n=2, d=2 → n=d → horizontal asymptote, not slant. ✗
II
g(x) = x²+4x+12x³+x²+2: n=2, d=3 → n<d → HA y=0, not slant. ✗
III
h(x) = x²+3x+22x+4: n=2, d=1 → n=d+1 ✓ → slant. Leading ratio = 12. Slant parallel to y=12x ✓
IV
k(x) = x⁴+x³+52x²+x−1: n=4, d=2 → n=d+2 → NOT slant (need exactly n=d+1). ✗
Answer: C) III only — h(x) has slant asymptote parallel to y = 12x ✓
⚡
Quick Reference
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🔑 Decision Tree
n = d → HA: y = ab
Same degree → ratio of leading coefficients
n < d → HA: y = 0
Denominator dominates → approaches zero
n = d+1 → Slant: y = abx
Exactly 1 more → slant asymptote
n > d+1 → No asymptote
Numerator dominates → end behavior like polynomial
⚠️ Common Mistakes
❌ HA = ratio of ALL coefficients
Only the leading coefficients matter for end behavior. Ignore all other terms.
❌ n>d always means slant asymptote
Slant requires exactly n=d+1. If n=d+2 or more, there is no horizontal or slant asymptote.
❌ Sign doesn't matter for limits
In Case 3, always check the sign of ab when substituting x→−∞ — a negative leading coefficient flips the sign.
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