Infinity Bound

Teacher Facilitation Guide

Subject: 12th Grade Calculus / Pre-Calculus

Topic: Limits & Asymptotes

Duration: 45 Minutes

Learning Objectives

Define and utilize limit notation for values approaching infinity and specific vertical asymptotes.
Numerically verify vertical and horizontal asymptotes using tables of values.
Analyze how transformations (specifically 'k') relate to the limit at infinity.

Required Materials

Classroom Slide Deck

Investigation Guide (1 per student)

Scientific Calculator Reference

Student Scientific Calculators

Pacing

5m

Warm-up: Limit Notation

10m

Video: Transformation Focus

25m

Main: Asymptote Investigation

5m

Closure: The 'k' Connection

Instructional Flow

1. Warm-up (5 min)

Introduce the notation \(\lim_{x \to \infty} f(x) = L\). Ask students: "If a function approaches a line but never touches it as \(x\) gets huge, what does that look like on a graph?" This bridges the gap between horizontal asymptotes and limit notation.

2. Video Viewing (10 min)

Watch the provided video. The key section is 0:23-2:21 where \(h\) and \(k\) are defined as vertical and horizontal asymptotes.

Pro Tip: Pause at 2:21. Ensure students have recorded the final general form \(y = \frac{a}{x-h} + k\).

3. Asymptote Investigation (25 min)

Distribute the Investigation Guide. Students will use the final equation from the video: \[y = \frac{4}{x-3} + 6\]

Numerical Analysis: Students calculate values closer and closer to \(x=3\) to see \(y \to \pm\infty\).
Infinite Horizon: Students calculate huge values of \(x\) (100, 1000, 10000) to see \(y \to 6\).

4. Closure (5 min)

Ask: "Why does the fraction term \(\frac{4}{x-3}\) 'disappear' as \(x\) becomes infinite?" Guide them to the realization that \(\text{finite} / \text{infinite} = 0\), leaving only \(k\). This is the fundamental reason why \(k\) is the horizontal asymptote.

Investigation Guide Answer Key

Part 1: The Vertical Wall (\(x \to 3\))

x	2.9	2.99	2.999	3.001	3.01	3.1
f(x)	-34	-394	-3994	4006	406	46

As \(x \to 3^-\), \(f(x) \to -\infty\). As \(x \to 3^+\), \(f(x) \to \infty\).

Part 2: The Infinite Horizon (\(x \to \infty\))

x	100	1,000	10,000
f(x)	6.0412	6.0040	6.0004

As \(x \to \infty\), the value of \(f(x)\) approaches 6. This confirms \(y=6\) is the horizontal asymptote.

Infinity Bound

Infinity Bound Teacher Guide

Infinity Bound Slides

Asymptote Investigation Guide

Scientific Calculator Reference

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