Standard 1 : Limits and Continuity (Archived)



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Develop an understanding of the concept of limit by estimating limits graphically and numerically and evaluating limits analytically. Extend the idea of a limit to one-sided limits and limits at infinity. Use limits to define and understand the concept of continuity, decide whether a function is continuous at a point, and find types of discontinuities. Understand and apply continuity theorems.

General Information

Number: MAFS.912.C.1
Title: Limits and Continuity
Type: Cluster
Subject: Mathematics - Archived
Grade: 912
Domain-Subdomain: Calculus

Related Standards

This cluster includes the following benchmarks
Code Description
MAFS.912.C.1.1: Understand the concept of limit and estimate limits from graphs and tables of values.
MAFS.912.C.1.2: Find limits by substitution.
MAFS.912.C.1.3: Find limits of sums, differences, products, and quotients.
MAFS.912.C.1.4: Find limits of rational functions that are undefined at a point.
MAFS.912.C.1.5: Find one-sided limits.
MAFS.912.C.1.6: Find limits at infinity.
MAFS.912.C.1.7: Decide when a limit is infinite and use limits involving infinity to describe asymptotic behavior.
MAFS.912.C.1.8: Find special limits such as
MAFS.912.C.1.9: Understand continuity in terms of limits.
MAFS.912.C.1.10: Decide if a function is continuous at a point.
MAFS.912.C.1.11: Find the types of discontinuities of a function.
MAFS.912.C.1.12: Understand and use the Intermediate Value Theorem on a function over a closed interval.
MAFS.912.C.1.13: Understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval.


Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Lesson Plan

Name Description
Leap Frog Review Game:

In this lesson students will demonstrate their knowledge of limits, graphing, and exact trig limits evaluated using substitution. The students will play a game in which they evaluate their knowledge of problems in the unit while it serves as a formative assessment for the teacher. The students receive immediate feedback on their work while the teacher works the problems, correcting errors or misconceptions. This lesson gives the student a power review of the concepts in the unit because the timing is determined by the teacher. All students are engaged and focused while playing this game. Giving students access to the PowerPoint of the game after the lesson provides a good study tool for the students.

Perspectives Video: Professional/Enthusiast

Name Description
Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Download the CPALMS Perspectives video student note taking guide.

Text Resource

Name Description
No Limits For Usain:

This informational text resource is intended to support reading in the content area. The text uses the story of Usain Bolt and his quest to reduce his world record in the 100 meter race to raise the question of whether there is a limit to his—or any human's—athleticism. The article uses number series, limits, and convergent and divergent series to prove that, hypothetically at least, a world record will go on reducing beyond any limit. A logistic curve is shown to model the data.

Tutorials

Name Description
Redefining a Function by Finding a Limit to Make the Function Continuous:

In this video students are introduced to an algebraic technique for rewriting a rational function in order to find a limit of the function.

Determine the Point a Function is Continuous:

This video will help you determine at which point a function is continuous.

Estimating Limit Numerically:

This video will help you determine how to estimate a limit.

Inferring Limit from Numerical Data:

This video will help you determine how to infer limits when given numerical data.

Limits at Positive and Negative Infinity:

In this video we will find the limit of a function as it approaches positive infinity and negative infinity.

More Limits at Infinity:

Here we will explore three more functions, and find their limits as they approach infinity.

Limits at Infinity Using Algebra:

Here we will use algebra to find the limit of a function with radicals as it approaches infinity.

Limits with Two Horizontal Asymptotes:

Here we will find the limit of a function as it approaches positive and negative infinity and results in two horizontal asymptotes.

Limit and Function Defined at Point of Discontinuity:

In this video we will determine if a limit exists at a point of discontinuity.

Limits to Define Continuity:

We will use limits to define continuity.

Limit by Factoring Cubic Expression:

Le'ts find a limit by factoring a cubic expression.

Limit Properties:

Let's look at some examples of limits using various properties.

Limits and Infinity:

We will look at examples of limits at infinity.

Example of a limit:

This video will demonstrate how to find a limit of a quotient. 

Introduction to limits:

This video will provide an introduction to the intuition behind limits through graphing.

How to determine a limit at a point of discontinuity:

This video demonstrates how to determine a limit at a point of discontinuity.

How to find One-sided limits from graphs:

You will be able to find one-sided limits from graphs.

Determining which limit statements are true:

This video demonstrates how to determine which limit statements are true.

Two-sided limit from a graph:

Determine the limit of a function from the left and from the right of an x-value from graph.



Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

Tutorials

Title Description
Redefining a Function by Finding a Limit to Make the Function Continuous:

In this video students are introduced to an algebraic technique for rewriting a rational function in order to find a limit of the function.

Determine the Point a Function is Continuous:

This video will help you determine at which point a function is continuous.

Inferring Limit from Numerical Data:

This video will help you determine how to infer limits when given numerical data.

Limits at Positive and Negative Infinity:

In this video we will find the limit of a function as it approaches positive infinity and negative infinity.

More Limits at Infinity:

Here we will explore three more functions, and find their limits as they approach infinity.

Limits at Infinity Using Algebra:

Here we will use algebra to find the limit of a function with radicals as it approaches infinity.

Limits with Two Horizontal Asymptotes:

Here we will find the limit of a function as it approaches positive and negative infinity and results in two horizontal asymptotes.

Limit and Function Defined at Point of Discontinuity:

In this video we will determine if a limit exists at a point of discontinuity.

Limits to Define Continuity:

We will use limits to define continuity.

Limit by Factoring Cubic Expression:

Le'ts find a limit by factoring a cubic expression.

Limit Properties:

Let's look at some examples of limits using various properties.

Limits and Infinity:

We will look at examples of limits at infinity.

Example of a limit:

This video will demonstrate how to find a limit of a quotient. 

Introduction to limits:

This video will provide an introduction to the intuition behind limits through graphing.

How to determine a limit at a point of discontinuity:

This video demonstrates how to determine a limit at a point of discontinuity.

How to find One-sided limits from graphs:

You will be able to find one-sided limits from graphs.

Determining which limit statements are true:

This video demonstrates how to determine which limit statements are true.

Two-sided limit from a graph:

Determine the limit of a function from the left and from the right of an x-value from graph.