Limit Logic Teacher Guide Teacher Guide: Limit Logic
Calculus Series
Limits & Continuity
Lesson Objectives
Distinguish between function values $f(a)$ and limit values $\lim_{x \to a} f(x)$.
Identify types of discontinuities (removable, jump, infinite, oscillating).
Understand the formal requirement for a limit to exist (LHL = RHL).
Instructional Sequence
5 MIN
Warm-up: The Arrival vs. The Destination
Students analyze two graphs in their worksheet. One has a filled hole (point exists at the path), and one has an empty hole (point is undefined or elsewhere). Use this to introduce the concept: "Limits are about the approach, not the arrival."
10 MIN
Video Investigation
Watch "Limits and Discontinuities" (Focus on 3:05 - 7:50). Focus on Graph B (Removable) vs. Graph C (Jump).
Key Discussion Point: Why does the limit exist at a hole but not at a jump? (Path alignment vs. Path break).
Video Link: https://www.youtube.com/watch?v=wX2omDL6iJQ
20 MIN
Activity: True/False/Fix
Students evaluate logic statements. For "False" statements, they must sketch a counter-example. This requires them to translate abstract notation into visual graphs.
10 MIN
Closure: The Great Debate
Pick the statement with the most disagreement (usually "If $f(a)$ is defined, the limit at $a$ must exist"). Have students defend their answers using their sketches.
Misconception Alert
Students often think "undefined" means the limit doesn't exist. Reinforce that removable discontinuities are the #1 proof that limits don't care about the point itself.
Quick Check
Ask: "If I'm walking toward a bridge that is out, can I still describe where I'm supposed to be going?" (The 'going' is the limit; the 'bridge out' is the undefined value).
Critical Vocabulary
Removable Jump Infinite Oscillating
Reference: Limits and Discontinuities (B)
Value vs Limit Anchor Chart Value vs. Limit
The "Arrival" vs. The "Approach"
f(a)
The Value
"Where are we at this exact moment?"
Represents the actual y-coordinate at $x=a$.
Can be undefined (a hole).
lim
The Limit
"Where do we expect to be based on the path?"
Requires Left-Hand approach = Right-Hand approach.
Doesn't care if the bridge is out ($f(a)$ undefined).
The Golden Rule of Limits
\(\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\)
If the roads don't meet, the limit is DNE!
Removable (Hole)
Limit EXISTS
The paths line up perfectly.
Jump
Limit DNE
The paths are broken.
Infinite
Limit DNE (or \(\pm \infty\))
Unbounded growth.
"A limit is a destination. You don't have to reach it to know where you're headed."
Limit Logic Lab Worksheet Limit Logic Lab
Topic: Function Values vs. Limit Existence
Student Name
Date
Warm-up: Analyze the Approach (5 min)
GRAPH A
1. Find $f(2)$:
2. Find $\lim_{x \to 2} f(x)$:
GRAPH B
1. Find $f(2)$:
2. Find $\lim_{x \to 2} f(x)$:
Reflect: What is the primary difference between these two scenarios regarding the "Limit" vs. the "Value"?
Video Discovery: Discontinuity Types (10 min)
Graph B: Removable
Commonly known as a "hole."
Observation:
Does the limit exist here? YES / NO
Graph C: Jump
Occurs in piecewise functions.
Observation:
Does the limit exist here? YES / NO
Activity: True / False / Fix (20 min)
Circle T or F . If the statement is False , provide a counter-example sketch in the box provided.
1
T F
"If $f(a)$ is undefined, then $\lim_{x \to a} f(x)$ cannot exist."
Counter-example Sketch
2
T F
"If $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = M$ (where $L \neq M$), the overall limit DNE."
Counter-example Sketch
3
T F
"If $f(a)$ is defined, then the limit at $x=a$ must exist and equal $f(a)$."
Counter-example Sketch
