Chaos at the Origin Worksheet Chaos at the Origin
AP Calculus AB • Limits & Discontinuities
NAME: _________________________________________________
DATE: _______________________ PERIOD: ________
WARM-UP: Sinusoidal Infinity
Consider the function \(f(x) = \sin(x)\). As \(x\) becomes arbitrarily large (\(x \to \infty\)), what happens to the output of the function? Does the limit exist?
My Prediction:
Sketch the behavior as \(x \to \infty\):
x y
VIDEO ANALYSIS: Oscillating Discontinuity
Clip: 7:50 - 10:25
1. Based on the video, why does \(y = \sin(1/x)\) fail to have a limit at \(x = 0\)?
2. What visual evidence from the "zoom" animation supports this conclusion?
ACTIVITY: Zoom into Chaos
Use Desmos or GeoGebra to graph each function. Zoom in repeatedly toward \(x = 0\). Record what happens to the \(y\)-values as \(x\) approaches zero from both sides.
Function Visual Behavior at \(x \approx 0\) Limit at \(x = 0\) \(y = \cos(1/x)\) DNE / ____ \(y = \sin(1/x) + x\)
| DNE / ____ |
| \(y = x \cdot \sin(1/x)\) |
| DNE / ____ |
THE SQUEEZE EFFECT
Compare the graphs of \(y = \sin(1/x)\) and \(y = x \cdot \sin(1/x)\). Why does the second one have a limit while the first one doesn't? Hint: Look at the amplitude as \(x \to 0\).
PRO-TIP: Persistent vs. Dampened Oscillation
Oscillation alone doesn't mean DNE. For a limit to fail, the oscillation must be persistent (never settling). If the function is "squeezed" toward zero as it shakes, the limit can still exist!
Chaos at the Origin Slides AP Calculus AB
CHAOS AT THE
ORIGIN
Investigating Oscillating Discontinuities & Limit Failures
Unit 1: Limits & Continuity
Warm-up
Consider the function:
As \(x \to \infty\), what happens to the output of the function?
Think-Pair-Share:
Does the limit exist? Why or why not?
\(x \to \infty\)
Formal Definition
A limit \(L\) exists at \(x = a\) if and only if:
\[\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\]
Hole
Limits Match
Jump
Limits Differ
Infinite
Unbounded Growth
Video Investigation
7:50 - 10:25
Embedded media
Watch For:
What happens to \(\sin(1/x)\) as we zoom in?
Does the graph ever settle on a value?
How is this different from a jump ?
Zoom into Chaos
Desmos / GeoGebra Challenge
Phase 1
Graph \(\cos(1/x)\) .
Compare it to the video's sine function.
Limit at 0?
Phase 2
Graph \(\sin(1/x) + x\) .
Does adding \(x\) fix the oscillation?
Approach 0
Phase 3
Graph \(x \cdot \sin(1/x)\) .
What happens to the "amplitude"?
Squeeze Time!
The Comparison
\(y = \sin(1/x)\)
Limit = DNE
The oscillation is persistent .
\(y = x \cdot \sin(1/x)\)
Limit = 0
The oscillation is dampened .
Final Takeaway
"Oscillation alone doesn't cause a limit to fail. It only fails if the amplitude remains non-zero as you approach the origin."
LHL = RHL?
Single Value?
Chaos at the Origin Answer Key Teacher Resource
Chaos at the Origin KEY
AP Calculus AB • Answer Key & Teaching Notes
WARM-UP: Sinusoidal Infinity
Sample Answer:
As \(x \to \infty\), \(\sin(x)\) continues to oscillate between -1 and 1. It never approaches a single value. Therefore, the limit does not exist (DNE) .
Sketch Guidance:
Graph should show persistent waves of constant amplitude (Height = 1)
VIDEO ANALYSIS: Oscillating Discontinuity
1. Why does \(y = \sin(1/x)\) fail to have a limit at \(x = 0\)?
As \(x\) gets closer to zero, the term \(1/x\) gets infinitely large. The sine of an infinitely large term oscillates between -1 and 1 faster and faster. Since it never settles on a single value, the limit fails.
2. What visual evidence from the "zoom" animation supports this conclusion?
The "Zoom Animation" shows the graph becoming more and more compressed. Even as we zoom in by factors of 10 or 100, the graph never looks like a line or a point; it looks like a solid block of color because the waves are so tight.
Activity Key: Zoom into Chaos
Function Behavior Description Limit Value \(y = \cos(1/x)\) Similar to sine, it oscillates infinitely fast between -1 and 1 as \(x \to 0\). DNE \(y = \sin(1/x) + x\) The oscillation is still persistent between \(\pm 1\) (slightly shifted by \(x\)). It doesn't settle. DNE \(y = x \cdot \sin(1/x)\) The amplitude of oscillation is controlled by \(x\). As \(x \to 0\), the waves are "squeezed" to zero height. 0
Teaching Tips
Misconception Alert
Students often think "oscillation = DNE" always. Emphasize that it's the amplitude that matters. If the amplitude doesn't approach zero, the limit doesn't exist.
Squeeze Theorem Intro
This is the perfect segue into the Squeeze Theorem. Point out that \(-|x| \leq x\sin(1/x) \leq |x|\). Since both \(-|x|\) and \(|x|\) go to 0, the function in the middle must too.
