Limit Mastery Station Slides
AP Calculus AB / BC • Unit 1 Intensive
4 Stations • 12 Minutes Each
LIMIT MASTERY
Review Stations Lab
Collaborative problem solving on limits, continuity, asymptotes, and AP-style analytical justifications.
Station 1
Graphs & Tables
One-Sided Limits
Station 2
Algebraic Forms
Conjugates & Trig
Station 3
Continuity & IVT
3-Part Test & Proofs
Station 4
Asymptotes
Infinite Limits
Mission Briefing
Station Norms & AP Guidelines
12 Min Work • 1 Min Rotate
1
Precision Notation
Write \(\lim_{x \to c}\) on every step until direct substitution occurs.
Dropped limits lose communication points!
2
No Fake Equivalence
Never write \(= \frac{0}{0}\). State numerator and denominator limits separately.
Writing "= 0/0" is mathematically invalid.
3
State The Hypotheses
Before applying IVT, explicitly write: \(f\) is continuous on \([a,b]\).
No continuity stated = zero IVT credit!
Every student records work on their station sheet. No Calculators Permitted
STATION 1
Graphical & Numerical Reasoning
Focus: One-Sided vs Two-Sided Limits
Function Profile Piecewise Behavior of \(g(x)\)
The function \(g\) is defined for all real numbers except \(x = 2\):
\(\lim_{x \to -1^-} g(x) = 4\)
\(g(-1) = 1\)
\(\lim_{x \to -1^+} g(x) = 1\)
\(\lim_{x \to 2^-} g(x) = +\infty\)
AP Criterion: \(\lim_{x \to c} g(x) = L\) if and only if \(\lim_{x \to c^-} g(x) = \lim_{x \to c^+} g(x) = L\).
Station Tasks
Task A: Does \(\lim_{x \to -1} g(x)\) exist? Justify using limit notation.
Task B: If \(h(x) = 3g(x) - [g(x)]^2\), evaluate \(\lim_{x \to -1^+} h(x)\).
Task C: Is \(g\) continuous at \(x = -1\)? State which continuity condition fails.
Discuss solutions together before recording on your master sheet.
Station 1 of 4 Active group collaboration required
Rotation Alert 60 Seconds Transition
Rotate to Station 2
Next Up: Algebraic Limits & Indeterminate Forms. Prepare to rationalize, apply trig limits, and solve for parameters!
AP Exam Trap To Avoid
Remember: \(g(c)\) has zero effect on whether \(\lim_{x \to c} g(x)\) exists. The limit only considers values approaching \(c\)!
STATION 2A
Algebraic Limits: 0/0 Techniques
Conjugates & Special Trig Limits
Problem 2.1 • Rationalizing Conjugate
\[ \lim_{x \to 4} \frac{\sqrt{x+5} - 3}{x - 4} \]
• Confirm direct substitution produces \(\frac{0}{0}\).
• Multiply numerator and denominator by \((\sqrt{x+5} + 3)\).
• Cancel the factor \((x - 4)\) before taking the final limit.
Problem 2.2 • Trigonometric Form
\[ \lim_{x \to 0} \frac{\sin(7x)}{3x} \]
• Recall theorem: \(\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1\).
• Rewrite expression as \(\frac{7}{3} \cdot \frac{\sin(7x)}{7x}\).
• Apply constant multiple limit law to evaluate.
Station 2 (Part 1 of 2) Keep limit operators until numbers are plugged in!
STATION 2B
Algebraic Limits: Parameter Challenge
High-Yield AP FRQ Skill
Problem 2.3: Finding Unknown Constants AP Exam Classic
Find the constants \(a\) and \(b\) such that:
\[ \lim_{x \to 2} \frac{x^2 + ax + b}{x - 2} = 5 \]
Step 1: Indeterminate Form Requirement
Because the denominator approaches 0 and the limit exists (equals 5), the numerator must also approach 0 as \(x \to 2\).
Step 2: Factoring & Solving
Factor \((x - 2)\) out of the numerator or use long division to equate coefficients and solve for \(a\) and \(b\).
Station 2 (Part 2 of 2) Verify your values of \(a\) and \(b\) by re-evaluating the limit!
Rotation Alert 60 Seconds Transition
Rotate to Station 3
Next Up: Continuity & The Intermediate Value Theorem. Master the 3-part continuity test and formal IVT proofs!
AP Justification Checklist
To prove continuity at \(x = c\), you must confirm all 3 parts: \(f(c)\) is defined, \(\lim_{x \to c} f(x)\) exists, and \(\lim_{x \to c} f(x) = f(c)\).
STATION 3A
Continuity at a Point: 3-Part Test
Piecewise Parameter Solving
Problem 3.1 • Solve for \(k\)
Find the value of \(k\) that makes \(f(x)\) continuous at \(x = 2\):
\[ f(x) = \begin{cases} kx^2 + 2x - 1, & x \le 2 \\ 5x - k, & x > 2 \end{cases} \]
Remember: Show both one-sided limits before setting them equal!
Required AP Written Proof
1. Left Limit: \(\lim_{x \to 2^-} f(x) = k(2)^2 + 2(2) - 1 = 4k + 3\)
2. Right Limit: \(\lim_{x \to 2^+} f(x) = 5(2) - k = 10 - k\)
3. Continuity: Set \(4k + 3 = 10 - k = f(2)\) and solve for \(k\).
Every student writes the full 3-step proof on their sheet.
Station 3 (Part 1 of 2) Verify your final value: \(5k = 7 \implies k = 7/5\)
STATION 3B
Intermediate Value Theorem (IVT)
AP FRQ Justification Template
Problem 3.2: Existence of a Root
Let \(g(x) = x^3 - 3x^2 + 4x - 5\). Explain why there must exist a number \(c \in (1, 3)\) such that \(g(c) = 0\).
1. Continuity Hypothesis
"Since \(g\) is a polynomial, \(g\) is continuous on \([1, 3]\)."
2. Evaluate Endpoints
\(g(1) = 1 - 3 + 4 - 5 = -3\) and \(g(3) = 27 - 27 + 12 - 5 = 7\).
3. State The Inequality
Since \(g(1) = -3 < 0 < 7 = g(3)\), zero lies between \(g(1)\) and \(g(3)\).
4. Conclude with IVT
"By the IVT, there exists at least one \(c \in (1, 3)\) such that \(g(c) = 0\)."
Station 3 (Part 2 of 2) Never skip stating that the function is continuous!
Rotation Alert 60 Seconds Transition
Rotate to Station 4
Final Station: Infinite Limits & Asymptotes. Look out for radical degrees and vertical asymptote directional signs!
Asymptote Definition Mastery
\(y = L\) is a horizontal asymptote if \(\lim_{x \to \infty} f(x) = L\) OR \(\lim_{x \to -\infty} f(x) = L\). Always check both directions!
STATION 4A
Limits at Infinity: The Radical Trap
Horizontal Asymptotes
Problem 4.1 • Find All Horizontal Asymptotes
\[ f(x) = \frac{\sqrt{9x^2 + 5}}{4x - 2} \]
Key Identity:
\(\sqrt{x^2} = |x| = x\) when \(x > 0\)
\(\sqrt{x^2} = |x| = -x\) when \(x < 0\)
End Behavior Breakdown
As \(x \to +\infty\):
\(\lim_{x \to \infty} \frac{\sqrt{9x^2}}{4x} = \frac{3x}{4x} = \frac{3}{4}\)
Horizontal Asymptote: \(y = 3/4\)
As \(x \to -\infty\):
\(\lim_{x \to -\infty} \frac{\sqrt{9x^2}}{4x} = \frac{-3x}{4x} = -\frac{3}{4}\)
Horizontal Asymptote: \(y = -3/4\)
Result: Two distinct horizontal asymptotes!
Station 4 (Part 1 of 2) Always write asymptote equations as lines: \(y = c\)
STATION 4B
Vertical Asymptotes vs Holes
Infinite One-Sided Limits
Rational Function Analysis
\[ m(x) = \frac{x + 3}{x^2 - 9} \]
Factored Form:
\(m(x) = \frac{x+3}{(x+3)(x-3)} = \frac{1}{x-3}\) for \(x \ne -3\)
Identify the removable discontinuity (hole) versus non-removable (vertical asymptote).
Station Tasks
Task A: Evaluate \(\lim_{x \to -3} m(x)\).
State coordinates of the hole.
Task B: Evaluate \(\lim_{x \to 3^-} m(x)\) and \(\lim_{x \to 3^+} m(x)\).
Determine if each approaches \(+\infty\) or \(-\infty\).
Task C: Write the equation of the vertical asymptote: \(x = 3\).
Ready for whole-class synthesis debrief!
Station 4 (Part 2 of 2) Final station complete • Regroup with whole class
Debrief & Synthesis
Master Synthesis Challenge
Whole-Class Final FRQ
AP Exam Synthesis Problem Combining All 4 Stations
\[ f(x) = \begin{cases} \frac{\sin(4x)}{2x}, & x < 0 \\ ax + b, & 0 \le x \le 3 \\ \frac{6x - 1}{x + 3}, & x > 3 \end{cases} \]
Question 1: Continuity at \(x = 0\) & \(x = 3\)
Find constants \(a\) and \(b\) that make \(f(x)\) continuous everywhere.
Question 2: End Behavior Asymptote
Evaluate \(\lim_{x \to \infty} f(x)\) and state the horizontal asymptote equation.
Submit your completed Station Record Sheet with full analytical justifications. Unit 1 Complete!
