Introduction to Limits Graphically

Active learning helps students move from static images to dynamic reasoning about how functions behave near specific points. For limits, this means wrestling with the tension between what the function 'does' and what it 'says' at a point, which is best explored through multiple representations and peer discussion.

• Grade 12
• Mathematics
• 3 activities
• 20 min–40 min

Learning Objectives

1. Analyze graphical representations of functions to determine the limit as x approaches a specific value from the left and right.
2. Compare the limit of a function at a point with the actual value of the function at that point, identifying differences.
3. Predict whether a limit exists at a given x-value based on the graphical behavior of the function, including the presence of holes or jumps.
4. Explain the graphical conditions under which a limit fails to exist at a specific x-value, such as oscillations or discontinuities.

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Paradox of the Hole

Students look at a graph with a hole at x=2. They discuss: 'What is f(2)?' versus 'What is the limit as x approaches 2?' Pairs then explain to the class why the limit can exist even if the point does not.

Prepare & details

Analyze how a function can have a limit at a point where the function itself is undefined.

Facilitation Tip: During the Think-Pair-Share, circulate and listen for students who conflate 'the function is undefined' with 'the limit does not exist,' then pose a counterexample graph to their group.

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

Inquiry Circle

⏱ 35 min·Small Groups

Inquiry Circle: Limit Numerical Analysis

Groups are given functions that are undefined at a point. They use calculators to plug in values like 1.9, 1.99, 1.999 and 2.1, 2.01, 2.001 to 'see' the limit emerge numerically before solving it algebraically.

Prepare & details

Predict when a limit fails to exist at a specific x-value based on graphical evidence.

Facilitation Tip: For the Collaborative Investigation, assign each group a different function with a hole or asymptote so that the whole class can compare patterns in the numerical tables.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Gallery Walk

⏱ 40 min·Small Groups

Gallery Walk: Continuity Critique

Post various graphs around the room (some continuous, some with jumps, holes, or asymptotes). Students walk around and must write the formal limit notation for the behavior at specific points on each graph.

Prepare & details

Differentiate between the value of a function at a point and the limit of a function at that point.

Facilitation Tip: During the Gallery Walk, place a prompt at each station asking students to critique the continuity claim made by the previous group, using specific evidence from the graph.

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

Teaching This Topic

Teachers should emphasize that limits describe the neighborhood around a point, not the point itself, so avoid rushing to algebraic rules. Use informal language like 'zooming in' on a graph to help students visualize the behavior. Research shows students benefit from drawing limits by hand before using technology, as it builds intuition for how functions behave near discontinuities.

What to Expect

Students will articulate when a limit exists or fails to exist by analyzing graphs, tables, and equations, using precise language such as 'approaches from the left,' 'approaches from the right,' and 'limit exists.' They will connect the graphical behavior to the formal definition of a limit as a destination the function gets arbitrarily close to, regardless of the actual value at that point.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share: The Paradox of the Hole, watch for students who claim the limit does not exist because the function is undefined at the hole.

What to Teach Instead

Ask them to trace the graph with their finger from both sides to the hole, then ask, 'What height is the graph trying to reach?' Revisit this after the numerical investigation to confirm their observations.

Common Misconception

During Collaborative Investigation: Limit Numerical Analysis, watch for students who assume the limit is the value in the table at the point of interest.

What to Teach Instead

Have them extend the table to x-values closer to the hole, then ask, 'What pattern do you see in the outputs?' Guide them to recognize that the limit is the value the outputs are approaching, not the missing value itself.

Assessment Ideas

Quick Check

After Think-Pair-Share: The Paradox of the Hole, display a graph with a hole and a jump. Ask students to write the left-hand limit, right-hand limit, and overall limit at the hole, then share with a partner to compare answers.

Discussion Prompt

During Collaborative Investigation: Limit Numerical Analysis, ask groups to present their findings on a function with a limit at x=a but f(a) undefined. Then pose, 'How can a function have a destination if the starting point is missing?' Have students explain using their numerical tables and graphs.

Exit Ticket

After Gallery Walk: Continuity Critique, give each student a graph with a vertical asymptote or jump. Ask them to sketch the graph, label the x-value where the limit fails, and write one sentence explaining why the limit does not exist, referencing the behavior from both sides.

Extensions & Scaffolding

• Challenge: Provide a piecewise function with a removable discontinuity and a jump. Ask students to find all x-values where the limit exists but the function is discontinuous.
• Scaffolding: Give students a partially completed table of values near a hole and ask them to complete it, then predict the limit.
• Deeper: Introduce the epsilon-delta definition informally by having students describe 'how close' x must be to a point to guarantee f(x) is within a small distance of the limit.

Key Vocabulary

Limit	The value that a function 'approaches' as the input (x) approaches some value, observed by looking at the graph from both sides.
One-sided limit	The value a function approaches as the input (x) approaches a specific value from either the left (lower values) or the right (higher values) only.
Discontinuity	A point on a graph where the function is not continuous, often appearing as a break, jump, or hole.
Hole (Removable Discontinuity)	A single point missing from a graph, indicated by an open circle, where the function is undefined but a limit may still exist.