Formal Definition of Limits and Continuity

This topic demands precision, and active learning lets students wrestle with the epsilon-delta machinery directly. By moving, sketching, and debating, they internalize why limits and continuity matter before formalizing proofs.

• Year 12
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

1. Analyze the epsilon-delta definition of a limit to prove the limit of simple polynomial and rational functions.
2. Justify the three conditions required for a function to be continuous at a specific point.
3. Classify discontinuities as removable, jump, or infinite based on limit behavior.
4. Construct a piecewise function that exhibits a specific type of discontinuity at a given point.
5. Evaluate the continuity of a function at a point using the formal epsilon-delta definition.

Chalk Talk

⏱ 30 min·Pairs

Pair Proof Relay: Epsilon-Delta Proofs

Pairs work on proving lim(x→a) f(x) = L for given functions. Student A chooses delta for a given epsilon and sketches the inequality; Student B verifies and suggests improvements. Switch roles after two proofs, then share one with the class.

Prepare & details

Differentiate between removable, jump, and infinite discontinuities in a function's graph.

Facilitation Tip: During Pair Proof Relay, circulate and listen for students to verbalize how delta depends on epsilon before writing symbols.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

Gallery Walk

⏱ 45 min·Small Groups

Gallery Walk: Discontinuity Classification

Groups receive graphs of functions with discontinuities. They label types (removable, jump, infinite), justify using epsilon-delta, and post on walls. Groups walk the gallery, adding peer feedback and corrections.

Prepare & details

Justify the conditions required for a function to be continuous at a point.

Facilitation Tip: While Small Group Graph Gallery Walk, require each group to post one classification decision with a one-sentence justification on the wall before rotating.

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

Chalk Talk

⏱ 25 min·Whole Class

Whole Class Example Builder: Continuity Exceptions

Class brainstorms functions continuous everywhere except one point. Vote on submissions, then test collectively with epsilon-delta checks projected on screen. Refine the winning example as a class.

Prepare & details

Construct an example of a function that is continuous everywhere except at a single point.

Facilitation Tip: In Whole Class Example Builder, only accept student examples that explicitly name the function, the point, and the three continuity conditions.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

Chalk Talk

⏱ 20 min·Individual

Individual Delta Slider Exploration: Limit Verification

Students use online applets to adjust epsilon and find corresponding delta for limits. Record three examples in journals, noting patterns for polynomials versus rationals.

Prepare & details

Differentiate between removable, jump, and infinite discontinuities in a function's graph.

Facilitation Tip: For Individual Delta Slider Exploration, set the slider’s minimum step size to force careful choices of delta and prevent guessing.

Setup: Large papers on tables or walls, space to circulate

Materials: Large paper with central prompt, Markers (one per student), Quiet music (optional)

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Self-Management

Teaching This Topic

Teachers should start with visual intuition before formalism. Draw a hole and ask students to estimate the limit informally, then revisit the same graph after proving with epsilon-delta. Emphasize that continuity is a three-part test, not just a limit. Research shows students grasp the definition better when they first experience the “game” of shrinking epsilon and finding delta interactively.

What to Expect

Students will confidently state the formal definitions, select appropriate delta for given epsilon values, and classify discontinuities by inspecting both graphs and analytic expressions. Success shows in clear justifications and accurate proof structures.

Watch Out for These Misconceptions

Common Misconception

During Pair Proof Relay, watch for students who assume f(a) equals the limit L without checking the function’s value at a.

What to Teach Instead

Require pairs to state f(a) explicitly, then test whether the limit equals that value; if not, classify the discontinuity type.

Common Misconception

During Small Group Graph Gallery Walk, watch for students who treat jump and infinite discontinuities as the same because both ‘blow up’ visually.

What to Teach Instead

Ask each group to measure or estimate the one-sided limits numerically and compare their finiteness, using graph scales and tables before classifying.

Common Misconception

During Individual Delta Slider Exploration, watch for students who always set delta equal to epsilon without analyzing the function’s slope or behavior.

What to Teach Instead

Prompt students to adjust the slider repeatedly while keeping epsilon fixed, then ask them to describe how delta changes relative to the function’s rate of change.

Assessment Ideas

Quick Check

After Small Group Graph Gallery Walk, hand each student a summary sheet with three unlabeled graphs and ask them to classify each discontinuity and state the limit if it exists.

Discussion Prompt

During Whole Class Example Builder, pose the prompt: ‘Can the limit exist at a point where the function is undefined?’ Have pairs debate, then build an example on the board to justify their claim using the formal definition.

Peer Assessment

After Pair Proof Relay, have each pair exchange their written proof with another pair. Reviewers score clarity, correct delta selection, and logical flow, then return feedback using a simple rubric.

Extensions & Scaffolding

• Challenge: Ask students to construct a piecewise function that is continuous everywhere except at one removable discontinuity, then prove the limit exists at that point.
• Scaffolding: Provide pre-labeled graphs with removable, jump, and infinite discontinuities and supply partial epsilon-delta outlines to fill in.
• Deeper exploration: Have students research the epsilon-delta definition of sequential limits and present how it compares to the traditional definition.

Key Vocabulary

Limit	The value that a function approaches as the input approaches some value. It describes the behavior of the function near a point, not necessarily at the point itself.
Epsilon-Delta Definition	A rigorous mathematical definition stating that for any arbitrarily small positive number epsilon, there exists a positive number delta such that if the input is within delta of the target value, the output is within epsilon of the limit.
Continuity at a Point	A function is continuous at a point 'a' if three conditions are met: the limit as x approaches 'a' exists, the function value f(a) is defined, and the limit equals f(a).
Discontinuity	A point where a function is not continuous. This can occur if the limit does not exist, if f(a) is undefined, or if the limit does not equal f(a).
Neighborhood	An open interval around a point. In the epsilon-delta definition, delta defines the 'x-neighborhood' and epsilon defines the 'y-neighborhood'.