Introduction to Limits

Active learning helps students grasp limits because the concept is counterintuitive and visual. When students manipulate tables, graphs, and sliders themselves, they see how values approach a target without necessarily reaching it. This hands-on engagement builds intuition before formal definitions take hold.

• 11th Grade
• Mathematics
• 4 activities
• 15 min–30 min

Learning Objectives

1. Analyze graphical representations of functions to estimate the limit as x approaches a specific value.
2. Calculate numerical approximations of a function's limit by constructing tables of values.
3. Compare the behavior of a function at a point with its behavior near that point to differentiate between the function's value and its limit.
4. Predict the behavior of rational functions as the input variable approaches positive or negative infinity based on their graphical and numerical trends.
5. Explain the concept of a limit using precise mathematical language, distinguishing between a limit and a function's value at a point.

Gallery Walk

⏱ 25 min·Pairs

Numerical Investigation: Getting Close Without Arriving

Pairs build tables of values for f(x) = (x squared minus 1) divided by (x minus 1) as x approaches 1 from both sides, using inputs like 0.9, 0.99, 0.999 and 1.1, 1.01, 1.001. They predict the limit, then factor and simplify to find the exact answer and explain why the function is undefined at x=1 but still has a limit.

Prepare & details

Explain what it means for a function to approach a limit.

Facilitation Tip: During Numerical Investigation, ask students to calculate f(x) for values getting closer to the target, then pause to ask what pattern they notice in the output.

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

Gallery Walk

⏱ 20 min·Small Groups

Graph Reading: Where Does the Function Go?

Groups receive graphs of piecewise and rational functions with holes, jumps, and asymptotes. For each graph, they identify the limit at specific points, the function value at those points, and whether the limit equals the function value. Groups annotate with limit notation and present their findings.

Prepare & details

Predict the limit of a function graphically and numerically.

Facilitation Tip: While students work on Graph Reading, circulate and ask them to trace the curve with their fingers as x approaches the target, emphasizing the direction of movement.

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: Value vs. Limit

Pairs are given three functions where f(a) is not equal to the limit of f(x) as x approaches a. Partners explain in their own words why these can differ, sketch examples of their own creation, and share with the class any cases where the distinction was surprising.

Prepare & details

Differentiate between a function's value at a point and its limit at that point.

Facilitation Tip: For Think-Pair-Share, provide a function with a hole at the limit point and ask students to articulate why the limit still exists even though f(a) is undefined.

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

⏱ 30 min·Small Groups

Desmos Exploration: Behavior Near a Point

Students graph functions on Desmos and zoom in near points of interest , holes, jumps, and corners , recording observations about what the function appears to be approaching. Small groups compare observations and discuss whether the function's value at the point matters for determining the limit.

Prepare & details

Explain what it means for a function to approach a limit.

Facilitation Tip: In Desmos Exploration, encourage students to zoom in around the point of interest to see how close the function gets to the limit value.

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

Teach limits by focusing on the language of approximation rather than precision. Use everyday examples like parking near a curb to illustrate getting close without touching. Avoid rushing to the formal epsilon-delta definition; instead, build comfort with the idea through repeated exposure to different representations. Research shows that students grasp limits better when they see them as a process rather than a static value.

What to Expect

Students will explain that a limit is about the behavior of a function near a point, not the value at the point. They will use tables, graphs, and verbal descriptions to justify their answers, showing they understand the difference between approaching and equaling. Misconceptions about continuity and undefined points will be addressed directly.

Watch Out for These Misconceptions

Common Misconception

During Numerical Investigation: Getting Close Without Arriving, watch for students who assume the output at the target x value is the limit.

What to Teach Instead

After they complete the table, ask them to compare f(a) with the values approaching a. Directly point out that the limit is the value they’re approaching, not the one at a, especially if f(a) is undefined or different from the limit.

Common Misconception

During Graph Reading: Where Does the Function Go?, watch for students who believe a limit requires the function to touch the y-value at the target x.

What to Teach Instead

Have students focus on the curve’s behavior near the point, not at it. Ask them to trace the path with their fingers and explain why the hole or gap doesn’t prevent the function from approaching the limit.

Assessment Ideas

Quick Check

After Graph Reading, provide students with a graph of a piecewise function and ask them to identify the limit as x approaches a point where the function is discontinuous. Have them explain whether the limit exists and why.

Exit Ticket

After Numerical Investigation, give students a table of x values approaching a target and corresponding f(x) values for a given function. Ask them to state the value the function appears to be approaching and compare it to f(a), if defined.

Discussion Prompt

During Think-Pair-Share, pose the question: 'Can a function have a limit at a point where it is undefined?' Have students provide examples or reasons, then facilitate a brief class discussion to address the idea of removable discontinuities.

Extensions & Scaffolding

• Challenge: Ask students to create their own function with a removable discontinuity and determine its limit at that point.
• Scaffolding: Provide a partially filled table for Numerical Investigation and ask students to complete the missing values.
• Deeper exploration: Have students research and present on real-world applications of limits, such as in physics or economics.

Key Vocabulary

Limit	The value that a function or sequence 'approaches' as the input or index approaches some value. The limit of a function at a point is the value the function gets arbitrarily close to as the input gets arbitrarily close to that point.
Approaching Infinity	Describes the behavior of a function as the input variable increases or decreases without bound. This helps understand the end behavior of graphs.
One-Sided Limit	The value a function approaches as the input approaches a specific point from either the left side (values less than the point) or the right side (values greater than the point).
Limit Does Not Exist (DNE)	Occurs when the left-sided limit and the right-sided limit at a point are not equal, or when the function's behavior is unbounded near that point.