STAGE 3: TESTING OUR LIMITS

In Stage 2 we looked at limits from a distance. Stage 3 is devoted to a closer look. Our primary objective is to understand what it means to say that a given function f(x) has a limit of L at a point x = a. Our goal is to be able to translate back and forth between a wide variety of limit statements and the numerical, graphical, and symbolic representations of functions. Extensive techniques for calculating limits will not appear until Stage 4.

Limits form one boundary between calculus and other areas of mathematics. The concept of infinity forms another. As we describe asymptotes and long-term behavior of functions, we will formulate the concept of infinity in terms of limits.

The key to Stage 3 is the definition of limit. It makes the notion of limit consistent and permits us to use algebra to verify the truth of a limit statement.

Objectives

Demonstrate an understanding of limits by relating limit statements to graphical observations and numerical data.

Draw valid conclusions about the graph or values of a function from a limit statement. Sketch graphs with features corresponding to various limit statements.
Formulate limit statements which describe graphical features and the numerical values of functions.

Know the definition of limit and correctly use and interpret notation for limits.
- Present/identify appropriate usages of the quantifiers "sufficiently " and "arbitrarily" in response to specific limit statements.
- Investigate right-hand and left-hand limits.
- Describe asymptotes in terms of infinite limits.

LESSON
What To Do:
Go to the Onward and Upward area for a description of the recorded activities that you must complete in this Stage.

Stage 3 T/F-multiple choice Recorded Quiz
Stage 3 Recorded Essay Quiz
The Lesson and Practice pages will help you prepare for the graded activities and contain material upon which further Stages are based. There are two Practice Quizzes to help you prepare for the Recorded Quizzes.
Side Trail: There is a long side trail in the Lesson. This trail treats the rigorous definition of limit.

PRACTICE

ONWARD
and UPWARD

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Last modified on Saturday, 21-Sep-96 09:18:09 PDT.

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