Home›Lesson Plans›Mathematics›Precalculus · Grade 12

Introduction to Limits: The Idea Behind Calculus

Students meet the limit — the powerful idea of the value a function approaches — the concept that makes all of calculus possible, from instantaneous speed to the area under a curve.

Grade 12Limits50 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Explain the idea of a limit.
  • ✓Read limit notation.
  • ✓Explain one-sided limits.
  • ✓Know when a limit does not exist.
Essential Question

What is a function “heading toward” as you zoom in on a point — even if it never quite arrives? This one idea is the doorway to all of calculus.

0
Lesson Phases
0
Vocabulary Terms
0
Standards Aligned
0
Interactive Task
Limits Explorer · Interactive

The Limits Explorer

Project this and tap each idea. Have students build intuition for what a limit is and why it opens the door to calculus.

🎯 Explore the idea of a limit — tap eachTry it

The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Heading Toward

5 min

What is a function “heading toward” as you zoom in?

👩‍🏫 Teacher Moves

  • Zoom in on a point on a curve.
  • Ask what value it nears.
  • Set the goal: limits.

🎒 Student Actions

  • Watch the zoom.
  • Find the value.
  • Predict the idea.
2

I Do — The Limit Idea

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Define a limit intuitively.
  • Introduce lim notation.
  • Explain one-sided limits.

🎒 Student Actions

  • Learn the idea.
  • Learn the notation.
  • Learn one-sided.
3

We Do — Explore Together

13 min

Class uses the Limits Explorer.

👩‍🏫 Teacher Moves

  • Tap each idea.
  • Estimate limits from a table/graph.
  • Spot a limit that fails.

🎒 Student Actions

  • Tap each idea.
  • Estimate limits.
  • Spot failures.
4

You Do — On Your Own

12 min

Students evaluate limits.

👩‍🏫 Teacher Moves

  • Estimate limits from graphs/tables.
  • Check one-sided agreement.
  • Identify nonexistent limits.

🎒 Student Actions

  • Estimate limits.
  • Check both sides.
  • Spot failures.
5

Close — Close

5 min

One limit.

👩‍🏫 Teacher Moves

  • Give a limit to estimate.
  • State whether it exists.
  • Hand out the exit ticket.

🎒 Student Actions

  • Estimate it.
  • State existence.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School: Precalculus).

CCSS
F-IF.4

Interpret key features of graphs, including behavior near a point (prepares for calculus limits).

CCSS
F-IF.7

Graph functions and analyze their behavior.

CCSS
F-BF.4

Understand inverse and approaching behavior of functions.

CCSS
F-IF.6

Calculate and interpret the average rate of change of a function.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Approaching-value number line.
  • Sentence frame: “As x approaches ___, f(x) approaches ___.”
  • Use a table of values.

Support & Access

IEP / 504
  • Estimate from a simple table.
  • Focus on “what value it nears.”
  • Use a clear graph.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Evaluate limits algebraically.
  • Explore limits at infinity.
  • Connect a limit to instantaneous slope.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻The Limits Explorer
  • 📐Graphing calculators
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

limitthe value a function approachesapproachto get closer and closer toone-sided limita limit from the left or rightdoes not existwhen a limit has no single valuecontinuoushaving no breaks or jumpsderivativethe instantaneous rate of changeintegralthe area under a curvecalculusthe math of change and accumulation
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
In your own words, what is a limit?
The value a function approaches (gets closer and closer to) as x approaches a certain number.
QUESTION 2
What does the notation lim(x→5) f(x) = 12 mean?
As x approaches 5, the value of f(x) approaches 12.
QUESTION 3
When does a limit fail to exist at a point?
When the left and right sides approach different values, or the function grows without bound (shoots to infinity).

Going deeper? Limits and slope.

Have students explore how the limit of an average rate of change becomes the instantaneous slope (the derivative) — the heart of calculus. A printable limits sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Introduction to LimitsFlip
Card 1
Term
Tap to flip →
Meaning
0

Nice work!

Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Introduction to LimitsQuiz
Score: 0
1 / 6
Question 1
0%

Nice work!

Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Introduction to LimitsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: approach, calculus, continuous, derivative, does not exist, integral, limit, one-sided limit
  1. the value a function approaches
  2. the math of change and accumulation
  3. a limit from the left or right
  4. the instantaneous rate of change
  5. the area under a curve
  6. when a limit has no single value
  7. having no breaks or jumps
  8. to get closer and closer to

Part B · Show what you learned

  1. In your own words, what is a limit?
  2. What does the notation lim(x→5) f(x) = 12 mean?
  3. When does a limit fail to exist at a point?
Answer key — Part A: 1) limit · 2) calculus · 3) one-sided limit · 4) derivative · 5) integral · 6) does not exist · 7) continuous · 8) approach
Part B: 1) The value a function approaches (gets closer and closer to) as x approaches a certain number. 2) As x approaches 5, the value of f(x) approaches 12. 3) When the left and right sides approach different values, or the function grows without bound (shoots to infinity).