Introduction to Limits: Graphical and Numerical

Active learning works well for introducing limits because students need to physically observe and discuss patterns in tables and graphs before formalizing ideas with notation. Moving between numerical values and visual representations builds the intuition that limits describe approach behavior, not function values.

• 12th Grade
• Mathematics
• 4 activities
• 15 min–20 min

Learning Objectives

1. Analyze graphical representations of functions to identify the behavior of the function as the input approaches a specific value.
2. Compare the values of one-sided limits to determine if a two-sided limit exists at a point.
3. Predict the limit of a function at a given point by examining numerical data from a table of function values.
4. Explain why a limit can exist at a point where the function itself is undefined, using graphical or numerical evidence.

Think-Pair-Share

⏱ 20 min·Pairs

Numerical Investigation: Approaching from Both Sides

Students fill in tables of x-values approaching a target from the left and right for four different functions, then make conjectures about whether the limit exists and why. Comparing tables across function types surfaces the difference between removable discontinuities and jump discontinuities before any formal definition is introduced.

Prepare & details

Explain how a limit can exist even if the function is undefined at a specific point.

Facilitation Tip: For Numerical Investigation: Approaching from Both Sides, have students work in pairs to create their own tables for different functions to see how values stabilize near a point.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Gallery Walk

⏱ 20 min·Small Groups

Gallery Walk: Does the Limit Exist?

Post graphs around the room, each with a highlighted point. Students annotate each with their limit assessment and reasoning. Whole-class debrief addresses cases that produced disagreement, building shared precision in limit language before notation is formalized.

Prepare & details

Compare one-sided limits to the overall limit of a function.

Facilitation Tip: During Gallery Walk: Does the Limit Exist?, post a variety of graphs and require students to physically move to each one and write their conclusions on sticky notes.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: Undefined but Approachable

Students analyze f(x) = sin(x)/x near x=0 numerically before graphing it. Pairs discuss how a function can consistently approach a value it never actually reaches, then share their reasoning with the class as an introduction to the formal limit concept.

Prepare & details

Predict the limit of a function based on its graphical behavior near a point.

Facilitation Tip: In Think-Pair-Share: Undefined but Approachable, assign each pair a different function with a removable discontinuity to analyze before sharing with the class.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 20 min·Pairs

Desmos Exploration: One-Sided Limits

Students drag a movable point along curves with various discontinuities, recording left-hand and right-hand y-values as the point approaches a target x. They categorize each function by limit existence and build vocabulary for one-sided limits from their own observations.

Prepare & details

Explain how a limit can exist even if the function is undefined at a specific point.

Facilitation Tip: Use Desmos Exploration: One-Sided Limits to let students manipulate sliders and immediately see how left and right limits interact at a jump discontinuity.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teachers often start with numerical and graphical explorations because students need concrete evidence to trust the abstract concept of a limit. Avoid rushing to formal notation; instead, use repeated cycles of observing behavior, predicting limits, and checking predictions. Research shows that students grasp limits better when they first describe what they see in tables and graphs before connecting it to the epsilon-delta language.

What to Expect

Students will confidently explain that limits describe behavior near a point, not at the point itself, and that both sides must agree for a two-sided limit to exist. They will use tables and graphs to estimate limits and justify their reasoning with clear language.

Watch Out for These Misconceptions

Common Misconception

During Numerical Investigation: Approaching from Both Sides, watch for students who conclude that a limit cannot exist because f(a) is undefined.

What to Teach Instead

Use the tables they create to redirect: ask them to focus on the values approaching a rather than the value at a, highlighting that missing values do not prevent observation of stable trends.

Common Misconception

During Gallery Walk: Does the Limit Exist?, watch for students who assume the limit must equal the function value at the point.

What to Teach Instead

Direct them to compare the limit value with f(a) on the graphs provided, asking them to explain when these values match and when they do not.

Common Misconception

During Think-Pair-Share: Undefined but Approachable, watch for students who think a one-sided limit is enough for the overall limit to exist.

What to Teach Instead

Use the jump discontinuities in their examples to show how left and right limits must agree; have them sketch and compare both sides to see why both are required.

Assessment Ideas

Exit Ticket

After Gallery Walk: Does the Limit Exist?, provide students with a graph of a piecewise function with a hole at x=2. Ask them to write the values the function approaches from the left and right, and whether the overall limit exists at x=2, explaining their reasoning in complete sentences.

Quick Check

During Numerical Investigation: Approaching from Both Sides, present a table of values for f(x) where f(3) is undefined but values near x=3 are 4.9, 4.99, 5.01, 5.1. Ask students to share their estimated limit in a class poll and explain how the table supports their answer.

Discussion Prompt

After Think-Pair-Share: Undefined but Approachable, ask students to discuss: 'Can a function have a limit at a point where it is not defined? Use an example from the activity to support your answer and explain your thinking to the class.'

Extensions & Scaffolding

• Challenge students to create their own piecewise function with a hole and a jump, then trade with a partner to find limits and explain why they exist or do not exist.
• Scaffolding: Provide pre-labeled tables with missing values for students to complete before estimating limits, focusing on functions like f(x) = (x^2 - 4)/(x - 2).
• Deeper exploration: Ask students to research the formal definition of a limit and create a visual or metaphorical explanation of how it matches their graphical and numerical observations.

Key Vocabulary

Limit	The value that a function's output approaches as the input approaches some value. The function's value at that point does not need to be defined.
One-sided limit	The value that a function's output approaches as the input approaches some value from only one direction, either from the left (less than) or from the right (greater than).
Graphical behavior	The way a function's graph rises, falls, or levels off as the input variable changes, especially near a particular point of interest.
Numerical approximation	Estimating the value of a limit by evaluating the function at input values that are increasingly close to the target input value.