Asymptote & Limit Lab Worksheet
AP Precalculus CED Unit 1: Topics 1.2, 1.7, 1.8
Polynomial & Rational Analysis
End Behavior, Limit Notation, Domain & Horizontal Asymptotes
Name:
Date: Period:
AP Limit Notation:
End behavior is formally modeled by \(\lim_{x \to -\infty} f(x)\) and \(\lim_{x \to \infty} f(x)\).
Limits evaluate to \(L \in \mathbb{R}\), \(\infty\), or \(-\infty\).
Part 1: Polynomial End Behavior & Leading Term Analysis
Dominant Term / Degree & Leading Coefficient
For each polynomial, identify the degree, leading coefficient (L.C.), and evaluate the limit statements representing the end behavior.
1. \(p(x) = -3x^5 + 7x^3 - 4x + 9\)
Degree: ________ (Even/Odd)
Leading Coeff: ________ (Pos/Neg)
\(\lim_{x \to -\infty} p(x) =\)
\(\lim_{x \to \infty} p(x) =\)
2. \(q(x) = 2(x - 1)^2(3 - x)(x + 4)\)
Dominant Term: \(a_n x^n = \) ________
Total Degree: ________
\(\lim_{x \to -\infty} q(x) =\)
\(\lim_{x \to \infty} q(x) =\)
3. Constructing Polynomials from Given End Behavior
A polynomial function \(g\) satisfies \(\lim_{x \to -\infty} g(x) = \infty\) and \(\lim_{x \to \infty} g(x) = \infty\), with real zeros only at \(x = -2\) (multiplicity 2) and \(x = 5\) (multiplicity 1).
Explain why the degree of \(g\) must be at least 4:
Write a possible formula for \(g(x)\) in factored form:
\(g(x) =\)
Part 2: Domain & Singularities of Rational Functions
Factoring & Interval Notation
State the domain of each rational function in interval notation. Identify all excluded values as either a vertical asymptote (V.A.) or a removable discontinuity (hole).
4. \(r(x) = \frac{x^2 - 9}{2x^2 + 5x - 3}\)
Factored form:
Domain:
V.A. at \(x =\)
Hole at \(x =\)
5. \(h(x) = \frac{3x - 6}{x^3 - 4x}\)
Factored form:
Domain:
V.A. at \(x =\)
Hole at \(x =\)
6. \(k(x) = \frac{x + 5}{x^2 + 16}\)
Denominator zeros:
Domain:
V.A. at \(x =\)
Hole at \(x =\)
AP Precalculus • Topic 1.2 & 1.7 Student Practice Page 1 of 2
Rational Functions & End Behavior
Horizontal Asymptotes, Degrees of Polynomials, and AP Synthesis
Unit 1: Topic 1.8 & 1.9
AP Precalc End Behavior Theorem for Rational Functions \(R(x) = \frac{p(x)}{q(x)}\):
\(\text{deg}(p) < \text{deg}(q)\): \(\lim_{x \to \pm\infty} R(x) = 0\) → H.A. \(y = 0\)
\(\text{deg}(p) = \text{deg}(q)\): \(\lim_{x \to \pm\infty} R(x) = \frac{a_n}{b_n}\) → H.A. \(y = \frac{a_n}{b_n}\)
\(\text{deg}(p) > \text{deg}(q)\): \(\lim_{x \to \pm\infty} R(x) = \pm\infty\) → No H.A.
Part 3: Horizontal Asymptotes & End Behavior Limits
Dominant quotient analysis
| Rational Function | Degree Comp. / Ratio of Leading Terms | \(\lim_{x \to \infty} f(x)\) | Horizontal Asymptote |
|---|
| 7. \(f(x) = \frac{8x^3 - 5x + 2}{2x^3 + 9x^2 - 1}\) | \(\frac{8x^3}{2x^3} =\) | | \(y =\) |
| 8. \(m(x) = \frac{5x^2 - 11}{3x^4 - 2x + 7}\) | \(\text{deg}(\text{num}) =\) _____ , \(\text{deg}(\text{den}) =\) _____ | | \(y =\) |
| 9. \(w(x) = \frac{(2x - 3)(5 - x)}{(x + 4)(3x - 1)}\) | Dominant term quotient: | | \(y =\) |
| 10. \(j(x) = \frac{-4x^3 + 6x}{x^2 - 5x + 6}\) | \(\text{deg}(\text{num}) > \text{deg}(\text{den})\) (unbounded) | | |
Part 4: AP Exam-Style Analytical Reasoning
Parameter Determination & Justification
11. Parameter Determination from End Behavior & Asymptotes
Let \(R(x) = \frac{a x^2 + bx - 12}{3x^2 - 12}\), where \(a\) and \(b\) are non-zero real constants. \(R\) satisfies: (i) horizontal asymptote at \(y = -4\), and (ii) removable discontinuity (hole) at \(x = 2\).
(a) Determine the value of \(a\). Justify using limit notation as \(x \to \infty\).
\(a = \) ________
(b) Determine the value of \(b\). Justify using properties of removable discontinuities.
\(b = \) ________
12. Complete Analytical Profile
Given \(g(x) = \frac{2x^2 - 8}{x^2 - 3x - 10} = \frac{2(x - 2)(x + 2)}{(x - 5)(x + 2)}\), fill in the complete analytical profile:
Domain:
Vertical Asymptote:
Removable Disc. (Hole):
Horizontal Asymptote:
AP Precalculus • Topic 1.8 & 1.9 End Behavior & Domain Analysis Page 2 of 2
