Infinite Horizons Worksheet
AP Precalculus • Topic 1.3 & 1.8
Infinite Horizons Worksheet
Polynomial End Behavior, Rational Horizontal Asymptotes & Limits at Infinity
Name:
Date: Period:
AP Precalculus Essential Concept Reference
Polynomial Leading Term: End behavior is dictated by \(a_n x^n\): • Even \(n\): same sign infinity as \(x \to \pm\infty\) • Odd \(n\): opposite signs as \(x \to \pm\infty\)
Rational End Behavior / H.A.: Let \(R(x) = \frac{a_n x^n + \dots}{b_m x^m + \dots}\): • \(n < m \implies \text{H.A. at } y = 0\) • \(n = m \implies \text{H.A. at } y = \frac{a_n}{b_m}\) • \(n > m \implies \text{No H.A. (unbounded)}\)
AP Limit Notation: Express end behavior strictly as:
\(\lim_{x \to -\infty} f(x) = L \quad \lim_{x \to \infty} f(x) = L\)
Part 1: Polynomial Functions & End Behavior Limit Notation State degree \(n\), leading coeff \(a_n\), and limits as \(x \to \pm\infty\)
- \(f(x) = -2x^4 + 7x^3 - 4x + 9\)
Degree \(n\): _____ Leading Coeff \(a_n\): _____
\(\lim_{x \to -\infty} f(x) =\)
\(\lim_{x \to \infty} f(x) =\)
- \(g(x) = 5x^5 - 3x^2 + 11x - 1\)
Degree \(n\): _____ Leading Coeff \(a_n\): _____
\(\lim_{x \to -\infty} g(x) =\)
\(\lim_{x \to \infty} g(x) =\)
- \(p(x) = (3 - x)(x + 2)^2(2x - 1)\)
Degree \(n\): _____ Leading Coeff \(a_n\): _____
\(\lim_{x \to -\infty} p(x) =\)
\(\lim_{x \to \infty} p(x) =\)
- \(q(x) = \frac{1}{3}x^2(6 - 2x^3)\)
Degree \(n\): _____ Leading Coeff \(a_n\): _____
\(\lim_{x \to -\infty} q(x) =\)
\(\lim_{x \to \infty} q(x) =\)
Part 2: Rational Functions • Horizontal Asymptotes & Limits Compare degrees, write H.A. equation (or "None"), and state limits
- \(r(x) = \dfrac{8x^2 - 3x + 1}{2x^2 + 5x - 7}\)
Deg Num: ___ | Deg Den: ___ H.A.: \(y =\)
\(\lim_{x \to -\infty} r(x) =\)
\(\lim_{x \to \infty} r(x) =\)
- \(h(x) = \dfrac{5x - 9}{3x^2 + 4}\)
Deg Num: ___ | Deg Den: ___ H.A.: \(y =\)
\(\lim_{x \to -\infty} h(x) =\)
\(\lim_{x \to \infty} h(x) =\)
- \(m(x) = \dfrac{-6x^3 + 4x^2 - 1}{2x^2 + 9}\)
Deg Num: ___ | Deg Den: ___ H.A.:
\(\lim_{x \to -\infty} m(x) =\)
\(\lim_{x \to \infty} m(x) =\)
- \(k(x) = \dfrac{(3x - 1)(4 - 2x)}{3x^2 + 8x + 2}\)
Deg Num: ___ | Deg Den: ___ H.A.: \(y =\)
\(\lim_{x \to -\infty} k(x) =\)
\(\lim_{x \to \infty} k(x) =\)
AP Precalculus • Topic 1.3 & 1.8 Practice Page 1 of 2
Infinite Horizons Worksheet | Page 2: Domain & Synthesis
Name:
Part 3: Domain of Rational Functions Set denominator \(\neq 0\); write domain in interval notation
- \(f(x) = \dfrac{2x + 7}{4x - 12}\)
Denominator restriction:
Domain: _________________________
- \(g(x) = \dfrac{x^2 - 3}{x^2 - 36}\)
Denominator restriction:
Domain: _________________________
- \(h(x) = \dfrac{3x - 1}{x^2 + 4x - 21}\)
Denominator restriction:
Domain: _________________________
- \(k(x) = \dfrac{5x^3 + 2}{x^2 + 25}\)
Denominator restriction:
Domain: _________________________
- \(p(x) = \dfrac{x + 4}{2x^2 - 5x - 3}\)
Denominator restriction:
Domain: _________________________
- \(q(x) = \dfrac{7 - x}{x(x^2 - 9)}\)
Denominator restriction:
Domain: _________________________
Part 4: AP Exam-Style Synthesis & Modeling Apply algebraic properties and limit definitions
Function Construction: Construct a possible rational function \(R(x) = \dfrac{a(x)}{b(x)}\) satisfying both conditions:
• Horizontal Asymptote: \(y = -3\) • Domain: \((-\infty, -2) \cup (-2, 5) \cup (5, \infty)\)
Equation: \(R(x) =\)
Let \(f(x) = \dfrac{12x^n - 5x + 2}{c x^3 + 7x^2 - 4}\), where \(n\) is a non-negative integer and \(c \neq 0\).
(a) If \(\lim_{x \to \infty} f(x) = 4\), find \(n\) and \(c\):
\(n =\) \(c =\)
(b) If \(n = 2\) and \(c = 3\), state H.A. & limit:
H.A.: \(y =\) \(\lim_{x \to -\infty} f(x) =\)
Consider polynomial \(P(x) = -2x^3 + 8x\) and rational function \(Q(x) = \dfrac{P(x)}{4x^3 + 1}\).
End behavior of polynomial \(P(x)\):
\(\lim_{x \to -\infty} P(x) =\)
\(\lim_{x \to \infty} P(x) =\)
Horizontal asymptote & limit of \(Q(x)\):
H.A.: \(y =\)
\(\lim_{x \to \infty} Q(x) =\)
AP Precalculus • Topic 1.8 & 1.10 Practice Page 2 of 2
