Limits and the Infinite

Active learning works for limits and the infinite because the abstract nature of infinity and unbounded behavior becomes tangible when students manipulate graphs, classify behaviors, and apply concepts to real data. Moving beyond symbolic manipulation to sorting, graphing, and context helps students build a deeper intuition that resists common oversimplifications about asymptotes and limits.

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

Learning Objectives

1. Calculate the limit of a function as x approaches infinity, describing the end behavior of rational functions.
2. Identify the location and behavior of vertical asymptotes by evaluating infinite limits at points of discontinuity.
3. Compare the end behavior of polynomial and exponential functions using limit notation.
4. Explain the role of limits at infinity in modeling population growth or decay over extended periods.
5. Analyze the convergence or divergence of sequences based on their limits as n approaches infinity.

Think-Pair-Share

⏱ 20 min·Small Groups

Sorting Activity: Infinite vs. Finite vs. No Limit

Cards describe limit scenarios in words, notation, and graphs. Groups sort them into three categories -- limit equals infinity (unbounded), limit equals a finite value, limit does not exist -- and justify each placement. A whole-class comparison identifies and resolves disagreements at the boundary cases.

Prepare & details

How can we describe the value a function approaches when that value is undefined?

Facilitation Tip: During the Sorting Activity, circulate and ask each pair to justify one card’s classification before moving to the next to ensure reasoning, not guessing, drives their choices.

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 End Behavior Hunt

Students graph a rational function and zoom progressively outward to identify horizontal asymptotes. They record limit notation for x going to positive and negative infinity, then compare results across different rational functions to identify the degree-comparison rule for end behavior.

Prepare & details

What distinguishes a jump discontinuity from a removable singularity in a physical system?

Facilitation Tip: In the Desmos End Behavior Hunt, pause students after each graph to share one observation about how the function’s degree or leading coefficient affects its horizontal asymptote.

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

⏱ 15 min·Pairs

Think-Pair-Share: Vertical Asymptote Analysis

Given f(x) = 1/(x-2), pairs write the left-hand and right-hand limits as x approaches 2, reconcile notation with the graph, and explain why the limit is positive infinity from one side and negative infinity from the other. They then generalize: what determines the sign of the infinite limit?

Prepare & details

Why is the concept of a limit necessary for understanding instantaneous change?

Facilitation Tip: For the Think-Pair-Share, assign roles so one student sketches the graph while the other writes the limit notation, ensuring both skills are practiced simultaneously.

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

Real-World Connections: Asymptotic Behavior in Context

Students examine a simplified pharmacology or population saturation graph where a quantity approaches a maximum as an input grows. They identify where asymptotic behavior becomes relevant and write limit statements describing the horizontal asymptote, connecting mathematical notation to a physically meaningful bound.

Prepare & details

How can we describe the value a function approaches when that value 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

Teaching This Topic

Teach this topic by starting with students’ informal experiences, then formalizing their language with precise notation. Avoid beginning with rules like L’Hôpital’s, which can obscure the geometric meaning of limits. Use sign charts early to connect algebraic behavior with graphical outcomes, and emphasize that infinity is a direction, not a destination. Research shows students grasp asymptotic behavior best when they contrast multiple examples that cross, approach from above or below, and have different end behaviors.

What to Expect

Successful learning looks like students confidently distinguishing finite limits, infinite limits, and no limits by connecting limit notation to graphs and real-world phenomena. They should articulate why infinity is not a number and how horizontal asymptotes describe long-term behavior rather than absolute boundaries.

Watch Out for These Misconceptions

Common Misconception

During the Sorting Activity: Watch for students labeling a limit equal to infinity as a finite value on the number line. Have them place ∞ beyond any marked real number to emphasize it is not a destination.

What to Teach Instead

Provide a number line with a clear break after large numbers; ask students to place ‘∞’ to the right of the last mark and explain why it sits there. Then return to their original cards and reclassify any infinite limits using this spatial understanding.

Common Misconception

During the Desmos End Behavior Hunt: Watch for students claiming that a function never touches its horizontal asymptote because it is a barrier. Redirect by asking them to graph a function that crosses y = 0 at x = 1 and observe its behavior as x grows.

What to Teach Instead

Ask students to adjust the constant term in their function to create a crossing point, then predict and verify the limit as x approaches infinity. Discuss why crossing does not violate asymptotic behavior.

Common Misconception

During the Think-Pair-Share: Watch for students assuming limits with denominator zero always go to infinity regardless of numerator sign. Provide a sign-chart template and ask them to evaluate the sign of (x+2)/(x-3) as x approaches 3 from both sides.

What to Teach Instead

Have students complete the sign chart together, then sketch the graph using their findings. Ask them to write the one-sided limits with correct signs to see how numerator behavior changes the result.

Assessment Ideas

Exit Ticket

After the Sorting Activity, give students the function f(x) = (2x^3 - x) / (x^3 + 5). Ask them to: 1. Calculate the limit as x approaches infinity. 2. Identify any horizontal asymptote. 3. Explain why their answers make sense based on the degrees of the polynomials.

Quick Check

During the Desmos End Behavior Hunt, after students have explored 5–6 graphs, display a new function and ask them to predict its horizontal asymptote as x approaches both positive and negative infinity. Collect responses via a show-of-hands or digital poll to identify misconceptions.

Discussion Prompt

After the Real-World Connections activity, pose the question: 'How would the long-term behavior of a drug concentration function change if the elimination rate doubled?' Facilitate a discussion using graphs or equations from the activity to assess understanding of horizontal asymptotes in context.

Extensions & Scaffolding

• Challenge students who finish early to design a function that has a horizontal asymptote at y = 3 but crosses it twice, then explain why their example works.
• For students who struggle, provide a set of pre-labeled graphs with blanks for limit notation; ask them to fill in the limits as x approaches ±∞ before writing their own.
• Deeper exploration: Ask students to research a real-world model (e.g., drug concentration over time) and explain how its horizontal asymptote represents a steady-state value, connecting to the Real-World Connections activity.

Key Vocabulary

Limit at Infinity	Describes the behavior of a function as the input variable x increases or decreases without bound. It indicates the horizontal or slant asymptote.
Infinite Limit	Describes the behavior of a function as the input variable x approaches a specific value, causing the output of the function to increase or decrease without bound. It indicates a vertical asymptote.
End Behavior	The behavior of a function's output as the input approaches positive or negative infinity. Often described by horizontal or slant asymptotes.
Vertical Asymptote	A vertical line x = c that the graph of a function approaches but never touches, occurring where the function's output approaches infinity or negative infinity.
Horizontal Asymptote	A horizontal line y = L that the graph of a function approaches as the input x approaches positive or negative infinity.