Special Limits Exit Ticket
Special Limits Speedway
Name: ____________________________________ Date: ___________
Exit Ticket
Apply the Special Limit rules learned in today's lesson to evaluate the following expressions. Show your reasoning if applicable.
Problem 01: The Constant
Evaluate:
\[ \lim_{x \to 7} 12 \]
Solution & Rule Applied:
Problem 02: The Power
Evaluate:
\[ \lim_{x \to -3} x^2 \]
Solution & Rule Applied:
Problem 03: The Root
Evaluate:
\[ \lim_{x \to 8} \sqrt[3]{x} \]
Solution & Rule Applied:
Pit Stop Check
Which of the four special limits do you find the most intuitive? Why?
Finish Line — Limit Laws Unit
Special Limits Speedway Slides
Special Limits Speedway
11th Grade Pre-Calculus: Defining foundational limit identities
Warm-Up: Track Inspection
Task 1: Graphing
Quickly sketch the following functions on a single coordinate plane:
\( y = 3 \) and \( y = x \)
Task 2: Prediction
Based on your graphs, what value does the function approach as \( x \to 2 \)?
A) \( \lim_{x \to 2} 3 = \text{?} \)
B) \( \lim_{x \to 2} x = \text{?} \)
The Pit Crew Briefing
Embedded media
Watch from 0:00 - 3:15. Focus on the four foundational rules.
Rules of the Road: Pt. 1
Rule 1: Constant
Limit of a Constant function \( c \)
\[ \lim_{x \to a} c = c \]
"No matter where you're going (\( a \)), a constant doesn't change!"
Rule 2: Identity
Limit of the function \( x \)
\[ \lim_{x \to a} x = a \]
"The limit simply goes where \( x \) is headed."
Rules of the Road: Pt. 2
Rule 3: Power
Limit of a power function \( x^n \)
\[ \lim_{x \to a} x^n = a^n \]
"Just 'plug in' the value being approached."
Rule 4: Root
Limit of a root function \( \sqrt[n]{x} \)
\[ \lim_{x \to a} \sqrt[n]{x} = \sqrt[n]{a} \]
"Valid for all \( a \) in the domain of the root."
Post-Race Analysis
Why do these make sense visually?
1
Look at the graph of \( y = 3 \). As you travel along the x-axis, does the y-value ever change? How does this explain \( \lim_{x \to a} c = c \)?
2
On the graph of \( y = x \), the y-value is always identical to the x-value. How does this connect to \( \lim_{x \to a} x = a \)?
Speed Limits!
Rapid-fire evaluation! I'll show a limit expression. You have 5 seconds to identify which rule applies and evaluate it.
Ready?
Set...
GO!
Lap 1: Easy Turns
5 SECONDS
Problem A
\[ \lim_{x \to 5} 10 \]
Problem B
\[ \lim_{x \to -2} x \]
Problem C
\[ \lim_{x \to 4} x^2 \]
Problem D
\[ \lim_{x \to 25} \sqrt{x} \]
Lap 2: The Hairpin
3 SECONDS
Problem E
\[ \lim_{x \to 0} \pi \]
Problem F
\[ \lim_{x \to -1} x^3 \]
Problem G
\[ \lim_{x \to 27} \sqrt[3]{x} \]
Problem H
\[ \lim_{x \to -100} x \]
Finish Line!
Great driving today. You've mastered the Special Limits.
Grab your Exit Ticket & Clear the Track!
Special Limits Teacher Guide
Teacher Guide
Special Limits Speedway
Unit: Limit Laws
Grade: 11th Pre-Calculus
Learning Objectives
- Identify the four special limit identities (Constant, Identity, Power, Root).
- Evaluate simple limits algebraically using these identities.
- Connect limit laws to visual/graphical representations of functions.
Lesson Pacing
Warm-up5 min
Video Instruction10 min
Class Discussion10 min
Speed Limits Activity15 min
Exit Ticket5 min
Facilitation: Discussion (10 min)
After the video, use these prompts to bridge the gap between "rules" and "graphs":
Rule 1 & 2 Focus
"If you are walking along the line y=3, and you look at your height (y-value) at x=2, x=5, or x=100, what do you notice? Does it ever change? Why does the limit of 3 always equal 3?"
Rule 3 & 4 Focus
"Wait... isn't this just direct substitution? Why can we do this for these functions but not for functions with holes or jumps? (Direct them toward 'continuity' without needing to name it yet.)"
Speed Limits Activity: Answer Key
A
10
Constant
B
-2
Identity
C
16
Power
D
5
Root
E
\(\pi\)
Constant
F
-1
Power
G
3
Root
H
-100
Identity
Caution Flags: Common Misconceptions
- • Constant Confusion: Students often try to plug \( a \) into the constant, e.g., saying \( \lim_{x \to 5} 10 = 50 \) or \( 5 \). Emphasize that a constant is a flat horizontal line.
- • The "Invisible" Rule: Students may evaluate correctly by intuition but struggle to name the rule. Encourage using formal names during the activity.
Proprietary Instructional Material — Speedway Series
