End Behavior Limits Worksheet
AP Precalculus • Topic 1.3
End Behavior & Limit Notation
Polynomial Dominance • Infinite Limits
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Date:
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AP Dominance Principle: As \(x \to \pm\infty\), \(P(x) = a_n x^n + \dots + a_0\) has the same limits as \(a_n x^n\).
\(\lim_{x \to \pm\infty} P(x) = \lim_{x \to \pm\infty} a_n x^n\)
1 \(f(x) = -3x^4 + 5x^3 - 2x + 7\) Standard Form
Leading Term & Degree
Term:
Degree: (Even / Odd)
Leading Coefficient
Value:
Sign: (\(+/- \))
Limit Statements
\(\lim_{x \to -\infty} f(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} f(x) = \underline{\hspace{1.6cm}}\)
2 \(g(x) = 4x^5 - 8x^3 + 11x - 1\) Standard Form
Leading Term & Degree
Term:
Degree: (Even / Odd)
Leading Coefficient
Value:
Sign: (\(+/- \))
Limit Statements
\(\lim_{x \to -\infty} g(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} g(x) = \underline{\hspace{1.6cm}}\)
3 \(h(x) = 14 - 6x - 2x^7\) Caution: Non-Ascending Order
Leading Term & Degree
Term:
Degree: (Even / Odd)
Leading Coefficient
Value:
Sign: (\(+/- \))
Limit Statements
\(\lim_{x \to -\infty} h(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} h(x) = \underline{\hspace{1.6cm}}\)
4 \(p(x) = -\frac{1}{2}x^2(x - 3)(x + 4)\) Factored Form
Leading Term Analysis
Product of leading factors:
Degree & Coefficient
Total Degree:
Leading Coeff:
Limit Statements
\(\lim_{x \to -\infty} p(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} p(x) = \underline{\hspace{1.6cm}}\)
5 \(k(x) = (2x - 1)^3(5 - x)\) Caution: Factor \((5 - x)\)
Leading Term Analysis
Compute \((2x)^3 \cdot (-x)\):
Degree & Coefficient
Total Degree:
Leading Coeff:
Limit Statements
\(\lim_{x \to -\infty} k(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} k(x) = \underline{\hspace{1.6cm}}\)
AP Precalculus • Topic 1.3 Polynomial Limits Practice Page 1 of 2
AP Precalculus: Polynomial Limits | Advanced Forms & Synthesis
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6 \(q(x) = \frac{3}{4}(x + 2)^2(x - 1)^3(x + 5)\) Factored with Multiplicities
Dominant Power
Sum of powers:
Leading term:
Parity & Sign
Degree is:
Coeff sign:
Limit Statements
\(\lim_{x \to -\infty} q(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} q(x) = \underline{\hspace{1.6cm}}\)
7 \(r(x) = -(3 - 2x)^3(x + 4)^2\) Caution: Negative Sign & Cube
Leading Term Analysis
\(- [(-2x)^3] \cdot [x^2] = \)
Degree & Leading Coeff
Degree:
Sign of \(a_n\):
Limit Statements
\(\lim_{x \to -\infty} r(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} r(x) = \underline{\hspace{1.6cm}}\)
8 \(m(x) = x^3(7 - 2x^2) + 5x^4 - 9\) Distributed Terms
Expanded Highest Degree
Distribute \(x^3(7 - 2x^2)\):
Identified Dominance
Highest Term:
Degree / Sign:
Limit Statements
\(\lim_{x \to -\infty} m(x) = \underline{\hspace{1.6cm}}\)
\(\lim_{x \to \infty} m(x) = \underline{\hspace{1.6cm}}\)
9 Root & Multiplicity Structural Representation AP Analytical Skill
A polynomial function \(w(x)\) has real roots at \(x = -4\) (mult. \(2\)), \(x = 0\) (mult. \(1\)), and \(x = 3\) (mult. \(3\)). No other roots exist. The graph passes through the point \((1, -100)\).
A. Leading Coefficient \(a\)
\(w(x) = a \cdot x(x+4)^2(x-3)^3\). Find sign of \(a\):
Degree \(n =\) Sign of \(a =\)
\(\lim_{x \to -\infty} w(x) = \underline{\hspace{1.8cm}}\)
\(\lim_{x \to \infty} w(x) = \underline{\hspace{1.8cm}}\)
10 AP Free-Response Reasoning & Transformations AP Conceptual Skill
Let \(v(x) = c x^n + \dots + a_0\) (\(c \neq 0\)). Given: \(\lim_{x \to -\infty} v(x) = -\infty\) and \(\lim_{x \to \infty} v(x) = \infty\).
A. Parity & Sign
Parity of \(n\) (Even / Odd):
Sign of \(c\) (\(+ / -\)):
B. For \(T(x) = -5 \cdot v(x)\):
\(\lim_{x \to -\infty} T(x) = \underline{\hspace{1.8cm}}\)
\(\lim_{x \to \infty} T(x) = \underline{\hspace{1.8cm}}\)
AP Precalculus • Topic 1.3 Polynomial Limits Practice Page 2 of 2
End Behavior Limits Answer Key
AP Precalculus • Topic 1.3 Teacher Answer Key
End Behavior Limits • Full Solutions
AP Scoring Rubric & Pitfalls
AP Exam Tip: Students must state limits using complete notation: \(\lim_{x \to -\infty} f(x) = \dots\) and \(\lim_{x \to \infty} f(x) = \dots\). Arrows alone (e.g. \(x \to \infty, y \to \infty\)) do not earn full notation points on AP Free-Response questions.
1 \(f(x) = -3x^4 + 5x^3 - 2x + 7\) ✓ Even Degree, Negative Coeff
Leading Term & Degree
Term: \(-3x^4\)
Degree: \(n = 4\) (Even)
Leading Coefficient
Value: \(a_4 = -3\)
Sign: Negative (\(a_n < 0\))
Correct Limits
\(\lim_{x \to -\infty} f(x) = -\infty\)
\(\lim_{x \to \infty} f(x) = -\infty\)
2 \(g(x) = 4x^5 - 8x^3 + 11x - 1\) ✓ Odd Degree, Positive Coeff
Leading Term & Degree
Term: \(4x^5\)
Degree: \(n = 5\) (Odd)
Leading Coefficient
Value: \(a_5 = 4\)
Sign: Positive (\(a_n > 0\))
Correct Limits
\(\lim_{x \to -\infty} g(x) = -\infty\)
\(\lim_{x \to \infty} g(x) = \infty\)
3 \(h(x) = 14 - 6x - 2x^7\) ⚠ Reorder: \(-2x^7 - 6x + 14\)
Leading Term & Degree
Term: \(-2x^7\)
Degree: \(n = 7\) (Odd)
Leading Coefficient
Value: \(a_7 = -2\)
Sign: Negative (\(a_n < 0\))
Correct Limits
\(\lim_{x \to -\infty} h(x) = \infty\)
\(\lim_{x \to \infty} h(x) = -\infty\)
4 \(p(x) = -\frac{1}{2}x^2(x - 3)(x + 4)\) ✓ Factored Analysis
Leading Term Analysis
\(-\frac{1}{2} \cdot (x^2) \cdot (x) \cdot (x) =\)
\(-\frac{1}{2}x^4\)
Degree & Coefficient
Degree: \(n = 4\) (Even)
Coeff: \(-\frac{1}{2}\) (\(a_n < 0\))
Correct Limits
\(\lim_{x \to -\infty} p(x) = -\infty\)
\(\lim_{x \to \infty} p(x) = -\infty\)
5 \(k(x) = (2x - 1)^3(5 - x)\) ⚠ Note: \((5 - x) \to -x\)
Leading Term Analysis
\((2x)^3 \cdot (-x) = 8x^3(-x) =\)
