AP Calculus Limits at Infinity Problem

AP Question of the Day Logistics and Context

• Title of Assessment: AP Question of the Day.

• Location: Link embedded within the Schoology calendar.

• Submission Window: Closes $5\,\text{min}$ after the start of class.

• Class Participants Present:

  • Offner, Noa

  • Nicotra, Sam…

  • Britton, Lesley M

  • KF

• Displayed Slide:

Problem 22 Statement and Answer Choices

• Question Text:

  Evaluate the following limit as $x$ approaches infinity:

  $\lim_{x \to \infty} \frac{-3x^2 + 7x^3 + 2}{2x^3 - 3x^2 + 5}$

• Answer Options:

  • Option A: $\infty$

  • Option B: $-\infty$

  • Option C: $1$

  • Option D: $\frac{7}{2}$

  • Option E: $-\frac{3}{2}$

Step-by-Step Mathematical Solution

• Step 1: Polynomial Term Rearrangement

  • Inspect the numerator: $-3x^2 + 7x^3 + 2$

  • Write the polynomial in standard form by arranging terms in descending order of power: $7x^3 - 3x^2 + 2$

  • Inspect the denominator: $2x^3 - 3x^2 + 5$

  • Confirm that the denominator is already arranged in standard form: $2x^3 - 3x^2 + 5$

• Step 2: Leading Degree Analysis for Limits at Infinity

  • Identify the degree of the numerator polynomial: $3$ (from term $7x^3$

  • Identify the degree of the denominator polynomial: $3$ (from term $2x^3$

  • Apply the rational function limit rule at infinity: when the degree of the numerator equals the degree of the denominator, the limit as $x \to \infty$ is equal to the ratio of the leading coefficients.

  • Leading coefficient of numerator: $7$

  • Leading coefficient of denominator: $2$

  • Computed Limit: $\frac{7}{2}$

• Step 3: Algebraic Verification via Division by Highest Power

  • Divide both numerator and denominator by $x^3$ (the highest power of $x$ in the denominator):

    $\lim_{x \to \infty} \frac{\frac{-3x^2}{x^3} + \frac{7x^3}{x^3} + \frac{2}{x^3}}{\frac{2x^3}{x^3} - \frac{3x^2}{x^3} + \frac{5}{x^3}}$

• Simplify each individual fractional term:

    $\lim_{x \to \infty} \frac{-\frac{3}{x} + 7 + \frac{2}{x^3}}{2 - \frac{3}{x} + \frac{5}{x^3}}$

• Evaluate the limit of each term as $x \to \infty$ using the fundamental property $\lim_{x \to \infty} \frac{c}{x^p} = 0$ for any constant $c$ and power $p > 0$:

    $\lim_{x \to \infty} \left(-\frac{3}{x}\right) = 0$

    $\lim_{x \to \infty} (7) = 7$

    $\lim_{x \to \infty} \left(\frac{2}{x^3}\right) = 0$

    $\lim_{x \to \infty} (2) = 2$

    $\lim_{x \to \infty} \left(-\frac{3}{x}\right) = 0$

    $\lim_{x \to \infty} \left(\frac{5}{x^3}\right) = 0$

• Substitute back to complete the limit calculation:

    $\frac{0 + 7 + 0}{2 - 0 + 0} = \frac{7}{2}$

Detailed Distractor and Common Error Analysis

• Option D (Correct Answer: $\frac{7}{2}$):

  • Represents the correct ratio of the leading coefficients $7$ and $2$ after recognizing that $7x^3$ is the highest degree term in the numerator.

• Option E (Distractor: $-\frac{3}{2}$):

  • Pitfall: Caused by looking strictly at the first term listed in the numerator ($-3x^2$) without reordering the polynomial into standard form first, taking $\frac{-3}{2}$ erroneously.

• Option C (Distractor: $1$):

  • Pitfall: Caused by incorrectly assuming matching leading coefficients or algebraic cancellation error.

• Options A and B (Distractors: $\infty$ and $-\infty$):

  • Pitfall: Caused by misidentifying the numerator as having a higher degree than the denominator, failing to recognize that both polynomials are cubic ($\text{degree} = 3$).