Mathematics · 11th Grade · Sequences, Series, and Limits · Weeks 28-36

Limits at Infinity and Asymptotes

Students will evaluate limits as x approaches infinity and relate them to horizontal asymptotes of functions.

About This Topic

Building on their introduction to limits, students now explore what happens to functions as x grows without bound. As x approaches positive or negative infinity, rational functions often settle toward a fixed horizontal value , a horizontal asymptote , or grow without bound. Evaluating these limits requires students to analyze the relative growth rates of the numerator and denominator, which comes down to comparing the degrees of the polynomials involved. This connects the algebraic study of rational functions to the language of limits, extending students' understanding of end behavior into a more formal framework.

Students learn three cases based on degree comparison: if the numerator's degree is less than the denominator's, the limit is 0; if the degrees are equal, the limit is the ratio of leading coefficients; if the numerator's degree exceeds the denominator's, the function grows without bound (no horizontal asymptote). This is a powerful synthesis topic, tying together polynomial behavior, rational functions, and the concept of limits.

Active learning with graphing tools is particularly effective here because students can test their predictions , computing the limit algebraically, then verifying with a graph or numerical table , within the same activity. Small-group discussion about why a rational function levels off at a specific value, connected to rate comparisons, builds durable understanding of asymptotic behavior.

Key Questions

1. Analyze the connection between limits at infinity and the presence of horizontal asymptotes.
2. Predict the end behavior of rational functions using limits at infinity.
3. Explain how limits at infinity describe the long-term behavior of a function.

Learning Objectives

• Evaluate the limit of a rational function as x approaches positive or negative infinity.
• Identify the horizontal asymptote of a rational function by analyzing its limit at infinity.
• Compare the degrees of the numerator and denominator of a rational function to predict its end behavior.
• Explain how the ratio of leading coefficients determines the value of a limit at infinity when degrees are equal.
• Calculate the limit of a rational function as x approaches infinity for cases where the numerator's degree is less than the denominator's.

Before You Start

Introduction to Limits

Why: Students need a foundational understanding of what a limit represents and how to evaluate basic limits before exploring limits at infinity.

Properties of Polynomials

Why: Understanding the degree and leading coefficients of polynomials is essential for comparing their growth rates.

Graphing Rational Functions

Why: Familiarity with the graphical behavior of rational functions, including intercepts and vertical asymptotes, provides context for horizontal asymptotes.

Key Vocabulary

Limit at Infinity	Describes the behavior of a function as the input variable x increases or decreases without bound, approaching positive or negative infinity.
Horizontal Asymptote	A horizontal line that the graph of a function approaches as x approaches positive or negative infinity. It represents the function's value at these extremes.
End Behavior	The behavior of a function's output values as the input values become very large (positive or negative).
Rational Function	A function that can be expressed as the ratio of two polynomial functions, where the denominator is not the zero polynomial.

Watch Out for These Misconceptions

Common Misconception

A function can never cross its horizontal asymptote.

What to Teach Instead

Unlike vertical asymptotes, horizontal asymptotes describe end behavior only , the function can cross the horizontal asymptote for finite x values. Students who learned 'never touch the asymptote' in earlier courses are often surprised. Graphical exploration in small groups that finds rational functions crossing their horizontal asymptote is the most direct correction.

Common Misconception

If the numerator's degree is bigger, there is simply no limit.

What to Teach Instead

The limit at infinity is plus or minus infinity in this case, which is informative end behavior describing unbounded growth. Students should say the limit does not exist as a finite number, or the limit is plus or minus infinity, rather than saying there is no limit at all. The distinction matters for precise mathematical communication.

Active Learning Ideas

Three-Case Investigation

Groups receive nine rational functions, three from each degree-comparison case. They compute limits at infinity algebraically, predict horizontal asymptote values, then graph each function to verify. Groups record their findings in a three-column summary and present one example from each case to the class.

25 min·Small Groups

Predict-Then-Verify with Desmos

Pairs receive five rational functions and must first predict the horizontal asymptote (or confirm none exists) using the degree comparison rule, then graph the function on Desmos and zoom out to verify end behavior. They annotate each graph with the computed limit and note any surprises.

25 min·Pairs

Think-Pair-Share: What Happens as x Gets Very Large?

Partners build a numerical table for a rational function at x equals 10, 100, 1000, and 10000, then predict the limit at infinity numerically before confirming algebraically. Discussion centers on whether the function ever actually reaches its horizontal asymptote.

15 min·Pairs

Gallery Walk: Match the Function to the Asymptote

Posters around the room each display a rational function. Students visit each poster and write the horizontal asymptote (or none) and the limit as x approaches infinity on a sticky note. A final comparison wall at the end of the gallery lets students resolve disagreements as a class.

20 min·Whole Class

Real-World Connections

• Environmental scientists use limits at infinity to model the long-term concentration of pollutants in a lake or the atmosphere, helping to set environmental regulations.
• Economists analyze the long-term growth trends of GDP or inflation using functions that approach limits at infinity, informing fiscal policy decisions.
• Engineers designing cooling systems for electronics might use limits at infinity to predict the stable operating temperature of a device as it runs continuously.

Assessment Ideas

Quick Check

Provide students with 3-4 rational functions. Ask them to calculate the limit as x approaches infinity for each and state if a horizontal asymptote exists, and if so, its equation. For example, 'Calculate lim x->inf (3x^2 + 1)/(x^2 - 5)' and 'Calculate lim x->-inf (2x + 4)/(x^2 + 1)'.

Discussion Prompt

Pose the question: 'Explain in your own words why the limit of f(x) = (x^2 + 2x)/(x^2 - 4) as x approaches infinity is 1, but the limit of g(x) = (x + 3)/(x^2 - 9) as x approaches infinity is 0.' Encourage students to discuss the role of the degrees of the polynomials.

Exit Ticket

On an index card, have students write down a rational function that has a horizontal asymptote at y = 2. Then, ask them to write one sentence explaining how they determined this asymptote based on the function's structure.

Frequently Asked Questions

What is a limit at infinity?

A limit at infinity describes what value (or behavior) a function approaches as x grows without bound. For example, the limit as x approaches infinity of (2x plus 3) divided by (x plus 1) equals 2, because as x gets very large, the constant terms matter less and less, and the ratio approaches 2 over 1.

How are limits at infinity related to horizontal asymptotes?

When a function has a finite limit at infinity, that limit value is the equation of the horizontal asymptote of the graph. A horizontal asymptote tells you where the function levels off for very large or very small x-values. If the limit is plus or minus infinity, there is no horizontal asymptote.

How do you evaluate a limit at infinity for a rational function?

Compare the degree of the numerator to the degree of the denominator. If the numerator's degree is smaller, the limit is 0. If the degrees are equal, the limit is the ratio of the leading coefficients. If the numerator's degree is larger, the function grows without bound and has no finite limit at infinity.

How does active learning strengthen understanding of limits at infinity?

End behavior is often easier to see than to derive. When students first predict asymptotes and then verify them graphically, the algebraic result becomes grounded in observable behavior. Small-group exploration that uncovers a rational function crossing its own horizontal asymptote challenges assumptions and produces discussions that build deeper understanding than formula application alone.