# STAGE  4:  AGENTLE  SLOPE

In Stage 3 we learned the concept and definition of limit. We have become good at guessing limits on the basis of numerical and graphical data. In this stage we will learn to compute limits exactly using algebra and properties of the Field Guide functions.

In many cases, the limit of a function at a point coincides with the value of the function at that point. This property is called continuity. Continuity is used in computing limits, but it is also the property needed to guarantee the existence of solutions to equations and to maximize or minimize quantities such as profits, volume, or times.

The related ideas of the limit and of continuity will complete our climb to the beginning of Differential Calculus proper. Understanding and computing limits is prerequisite to understanding and computing derivatives, while continuity lurks in almost all of the problems which differential Calculus is designed to solve.

Objectives

• Know the definition of continuity and test functions for continuity using graphs, tables, or algebra. Know where all the Field Guide functions are continuous.

• Use limit laws and continuity to compute exact values of limits.

• Understand the squeeze law for limits and be able to compute limits which involve the limit as x approaches zero of sin(x)/x.

• Know the statement and uses of the Intermediate Value Theorem. Know the statement of the Extreme Value theorem. Distinguish between valid and invalid applications of these theorems.

 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.

• Two T/F-multiple choice Quizzes (20 points each)
• Stage 4 Essay Quiz (20 points)
The Lesson and Practice pages will help you prepare for the graded activities and contain material upon which further Stages are based.

Side Trails: There are several optional side trails in the Lesson. These trails add precision and give more rigrous proofs of material covered on the core pages.

 PRACTICE
 ONWARD and UPWARD

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