What is End Behavior?
How a function behaves as x increases or decreases without bound
📐 End Behavior and Limit Notation
⬅️ Left End Behavior
lim f(x) as x → −∞
As the x-values decrease without bound, the y-values of f(x) approach…
➡️ Right End Behavior
lim f(x) as x → +∞
As the x-values increase without bound, the y-values of f(x) approach…
Reading End Behavior from Graphs — Example 1
Follow the arrows at the far left and far right of each graph
Graph a)
lim(x→−∞) f(x) = ∞
lim(x→∞) f(x) = −∞
Left: y → ∞ · Right: y → −∞
Graph b)
lim(x→−∞) f(x) = ∞
lim(x→∞) f(x) = ∞
Left: y → ∞ · Right: y → ∞
How to read end behavior from a graph
Look at the far left tail of the graph — is it pointing up (→ ∞) or down (→ −∞)? That's your left end behavior. Then look at the far right tail — same question. Write the limit statements for each.
The Two Rules for Polynomial End Behavior
Always determine the RIGHT side first, then use degree to get the LEFT side
➡️ RIGHT Side (x → ∞)
1
Goes to ∞ if the leading coefficient is positive.2
Goes to −∞ if the leading coefficient is negative.Think: the right side follows the sign of the leading coefficient.
⬅️ LEFT Side (x → −∞)
1
Goes the SAME direction as the right if the degree is EVEN.2
Goes the OPPOSITE direction as the right if the degree is ODD.Think: even = same, odd = opposite. (Like the parity of numbers!)
📊 All 4 Cases at a Glance
| Degree | Leading Coefficient | Left (x→−∞) | Right (x→+∞) | Example |
|---|---|---|---|---|
| Even | Positive | +∞ | +∞ | x², x⁴ |
| Even | Negative | −∞ | −∞ | −x², −2x⁶ |
| Odd | Positive | −∞ | +∞ | x³, 4x⁵ |
| Odd | Negative | +∞ | −∞ | −x³, −x⁵ |
Step-by-step method
Step 1: Find the leading term (highest power of x).
Step 2: Determine RIGHT side: positive LC → +∞, negative LC → −∞.
Step 3: Determine LEFT side: even degree → same as right, odd degree → opposite of right.
Determining End Behavior — Examples 3 & 4
Identify the leading term, apply the two rules, write limit statements
Odd · Positive LC
f(x) = 4x⁵
L: −∞
R: +∞
Degree 5 (odd), LC = 4 (pos). Right → +∞, Left → opposite = −∞
Even · Positive LC
g(x) = ½x⁴
L: +∞
R: +∞
Degree 4 (even), LC = ½ (pos). Right → +∞, Left → same = +∞
Even · Negative LC
y = −2(x+3)⁶
L: −∞
R: −∞
Degree 6 (even), LC = −2 (neg). Right → −∞, Left → same = −∞
Odd · Negative LC
h(x) = 3 − x⁵
L: +∞
R: −∞
Leading term −x⁵, degree 5 (odd), LC = −1 (neg). Right → −∞, Left → opposite = +∞
Odd · Negative LC
k(x) = 8x² + 4 − x⁵
L: +∞
R: −∞
Leading term −x⁵ (degree 5, LC = −1 neg). Right → −∞, Left → +∞
Even · Negative LC
m(x) = 2x(x−1)(6−x)
L: +∞
R: −∞
Expand: leading term = 2x · x · (−x) = −2x³. Degree 3 (odd), LC = −2 (neg). Right → −∞, Left → +∞
📌 Example 4 — Write full limit statements for end behavior.
f(x) = −3x⁴
Even degree · LC = −3 (negative)
Right → −∞ · Left → same (even) = −∞
lim(x→−∞) f(x) = −∞
lim(x→∞) f(x) = −∞
Right → −∞ · Left → same (even) = −∞
lim(x→−∞) f(x) = −∞
lim(x→∞) f(x) = −∞
g(x) = 5x³ + 2x² − 7
Odd degree · LC = 5 (positive)
Right → +∞ · Left → opposite (odd) = −∞
lim(x→−∞) g(x) = −∞
lim(x→∞) g(x) = ∞
Right → +∞ · Left → opposite (odd) = −∞
lim(x→−∞) g(x) = −∞
lim(x→∞) g(x) = ∞
End Behavior of Non-Polynomial Functions — Example 5
Quadratic, rational, and power functions — use a graphing calculator
Non-Polynomials: use your calculator
For rational functions like (2x−3)/(x+1) or functions like −10/x², the polynomial rules don't apply. Use a graphing calculator to trace the tails, or remember: rational functions with equal-degree numerator/denominator approach y = (ratio of leading coefficients) as a horizontal asymptote.
📌 Example 5 — Use a calculator to determine end behavior.
a
f(x) = x² − 3x + 1 — Polynomial, degree 2 (even), LC = 1 (positive) → both sides go to +∞.lim(x→−∞) = ∞ · lim(x→∞) = ∞
b
g(x) = (2x − 3)/(x + 1) — Rational function. As x → ±∞, the lower-order terms drop off: leading numerator 2x, leading denominator x → ratio = 2/1 = 2. Horizontal asymptote at y = 2.lim(x→−∞) = 2 · lim(x→∞) = 2
c
h(x) = −10/x² — As x → ±∞, the denominator x² grows without bound, so h(x) → 0 from below. Horizontal asymptote at y = 0.lim(x→−∞) = 0 · lim(x→∞) = 0
Quick Reference
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🔑 The 3-Step Process
Step 1: Find leading term (highest power)
Step 2: RIGHT side → LC pos = +∞, neg = −∞
Step 3: LEFT side → even=same, odd=opposite
Verbal: "As x → −∞, f(x) → ___"
Limit: lim(x→−∞) f(x) = ___
Limit: lim(x→−∞) f(x) = ___
⚠️ Common Mistakes
❌ Using the FIRST term written as leading term
Always find the HIGHEST power. In k(x) = 8x² + 4 − x⁵, the leading term is −x⁵, not 8x².
❌ Confusing even/odd with positive/negative
Even degree → SAME direction both ends. Odd degree → OPPOSITE directions. This has nothing to do with the sign of the LC.
❌ Not expanding factored polynomials
For m(x) = 2x(x−1)(6−x), you must find the leading term by multiplying highest powers: 2x · x · (−x) = −2x³.
❌ Applying polynomial rules to rational functions
For (2x−3)/(x+1), the end behavior is NOT ±∞ — it approaches the horizontal asymptote y = 2. Use a calculator or ratio of leading coefficients.