Infinity Blueprint Teacher Guide Infinity Blueprint
Teacher Facilitation Guide
Subject: Calculus / Precalculus
Topic: Infinite Limits
Learning Objective
Students will investigate infinite limits and the concept of "close enough" by analyzing function behavior near vertical asymptotes using numerical and graphical evidence.
Materials Needed
Laptops/Tablets with Desmos
"Zoom Into Infinity" Worksheet
"Infinite Behavior" Anchor Chart
Video: Introduction to Limits (7:01-8:48)
Lesson Procedure
5 MIN
The Hook: The Grey Box
Project the slide showing the "grey box" hiding the graph of \(g(x)\) near \(x = -3\). Ask students: "Based on what you see outside the box, what do you think the limit is?"
Teacher Note: Most students will guess 0.5 or 1 based on the visible slope. Do not reveal the answer yet.
10 MIN
Video Viewing & Discussion
Play the video from 7:01 to 8:48 . Discuss the revelation of the asymptote.
Why was 0.5 a bad guess? (We weren't "close enough")
What does "infinity" mean in the context of a limit?
Key Quote: "So close that they're practically brushing right up against a."
20 MIN
The "Zoom In" Activity
Students use Desmos to investigate three specific functions. They must record \(y\)-values at a distance of 0.0001 from the asymptote \(x\)-value.
TARGET FUNCTIONS:
f(x) = 1/(x-2)
g(x) = 1/(x+3)^2
h(x) = (x+1)/(x-4)
5 MIN
Reflection: How Close is Close?
Facilitate a class discussion. "In the video, why did the initial guess fail? When did the true behavior become obvious?" Focus on the idea that in calculus, "close" means an infinitely small distance.
FINISH
Formal Definition
Define Vertical Asymptote : An \(x\)-value where the limit of the function as \(x\) approaches that value is \(\infty\) or \(-\infty\).
Common Misconceptions
Infinity is a number: Remind students that \(\infty\) describes behavior (getting larger without bound), not a coordinate."At" vs "Near": Students often try to plug in the asymptote value (e.g., \(f(2)\)). Emphasize that we are checking \(2.0001\), not \(2\). Zoom Infinity Worksheet Zooming Into Infinity
Student Investigation: Limits & Asymptotes
Name:
Date:
1
The Hook: Initial Prediction
Look at the graph on the board. A grey box is hiding the behavior at x = -3.
What is your "far away" guess for \(\lim_{x \to -3} g(x)\)?
The Reveal: What was the actual behavior?
2
Video Viewing: Why "0.5" failed
The narrator mentioned that to find a limit, we must look at values that are:
Explain why looking at \(x = -2\) or \(x = -4\) (from "far away") was misleading for this function:
3
The "Zoom In" Challenge
Open Desmos. For each function, zoom in repeatedly at the asymptote until you are within 0.0001 units of the target value. Record the results below.
f(x) = 1 / (x - 2) Target: x = 2
Location x-value y-value (f(x)) Just Left 1.9999 Just Right 2.0001
g(x) = 1 / (x + 3)² Target: x = -3
Location x-value y-value (g(x)) Just Left -3.0001 Just Right -2.9999
h(x) = (x + 1) / (x - 4) Target: x = 4
Location x-value y-value (h(x)) Just Left 3.9999 Just Right 4.0001
4
Reflection: Close Enough?
When you zoomed in, what happened to the y-values compared to when you were at x = 0 or x = 10?
Look at Function B. As you approach \(x = -3\) from both sides, the y-values both go to positive infinity. Does this function have a Limit at \(x = -3\)? Why or why not?
The Blueprint Rule
A Vertical Asymptote exists at \(x = a\) if the function "explodes" to \(\pm\infty\) as you get closer and closer.
In your own words, what is the "close enough" rule for limits?
Infinite Behavior Anchor Chart Infinite Behavior
The "Close Enough" Rule
Vertical Asymptote
Official Notation
\[ \lim_{x \to a} f(x) = \infty \]
"As x gets closer and closer to 'a', the height of the function grows without bound."
WHAT IS "CLOSE ENOUGH"?
In Calculus, context is everything. If you are at \(x = 2.5\), you are too far! You must look at \(x = 2.0001\).
Distance < 0.0001
REMEMBER THE GREY BOX?
Don't guess from a distance. A function can look like it's approaching 0.5 when viewed from far away, but its true nature is revealed only when you get extremely close .
Key Takeaway
Limits = Proximity,
Infinite Behavior Slides Zooming Into Infinity
Understanding Function Behavior & Vertical Asymptotes
The Grey Box Challenge
Predict the Behavior
"Based on the puzzle pieces we can see, what is the limit of the function as x approaches the grey box?"
0.5 ?
1.0 ?
Let's Reveal the Truth
Time: 7:01 - 8:48
Embedded media
Watch closely as the "grey box" is removed. Why was our initial guess from far away so wrong?
The "Close Enough" Principle
The Problem
Looking from "far away" hides the true behavior of functions at asymptotes.
The Solution
To accurately find a limit, we must look at values very close to the target point.
Distance < 0.0001
Your Turn: Zoom In!
Desmos Exploration Activity
1
Launch Desmos
Enter the functions listed on your worksheet one by one.
2
Zoom & Table
Zoom in until you can see the behavior at exactly 0.0001 distance.
3
Record Data
Note the massive y-values. Is it going to \(\infty\) or \(-\infty\)?
Pro Tip: Use the 'Table' feature in Desmos to test specific x-values like 1.9999 and 2.0001!
