Chapter 2

Limits and Continuity

Chapter Overview

Connections

The yield on the June to August rice crop in Laos depends on the amount of rainfall, which is usually between 0.5 and 0.8 meter over this three-month growing season. More rain produces a higher yield, but the benefits diminish as the amount of rain increases. Crop yield in normal years can be approximated by

C(r) = -2.1 + 8.7r - 3.7r^2

metric tons per hectare, where r is the total rainfall, measured in meters. When the total rainfall is about 0.7 meter, how sensitive is the crop yield to a small increase in the amount of rain? Section 2.4 can help answer this question.

Chapter 2 Overview

The concept of limit is one of the fundamental building blocks of calculus, enabling us to describe with precision how change in one variable affects change in another variable. In this chapter, we show how to define and calculate limits of function values. The calculation rules are straightforward, and most of the limits we need can be found by substitution, graphical investigation, numerical approximation, algebra, or some combination of these.

One of the uses of limits lies in building a careful definition of continuity. Continuous functions arise frequently in scientific work because they model such an enormous range of natural behavior and because they have special mathematical properties.

Section 2.1: Rates of Change and Limits

Lesson Objectives

Determine average speed.
Compare average speed to instantaneous speed.
Define the limit of a function and the properties of limits.
Identify conditions under which a limit does and does not exist.
Determine one-sided and two-sided limits of functions.

Section 2.1

Complete the following exercises, beginning on page 65:

1–9 (odd problems only), 15, 21, 25, 27, 43, 45–49 (odd problems only), 55

Routine Practice

01 - Limits by Direct Evaluation.pdf

Limits by Direct Substitution

01 - Limits at Removable Discontinuities.pdf

Limits at Removable Discontinuity

#1-10 only

Section 2.2: Limits Involving Infinity

Lesson Objectives

Calculate limits as x goes to positive and negative infinity.
Find vertical and horizontal asymptotes using limits.
Determine end behavior of a function using limits.

Section 2.2

Complete the following exercises, beginning on page 76:

1, 3, 5, 7, 27, 29, 31, 53

Routine Practice

01 - Limits at Essential Discontinuities.pdf

Limits at Essential Discontinuities

01 - Limits at Infinity.pdf

Limits at Infinity

Quick Quiz: Sections 2.1 and 2.2

Section 2.3: Continuity

Lesson Objectives

Identify intervals of continuity and discontinuity over intervals of a function.
Identify types of discontinuity, including jump, infinite, and oscillating.
Modify or extend a function to remove discontinuities.
Use properties of continuous functions to determine function continuity over algebraic combinations.
Use the intermediate value theorem to verify continuity.

2.3: Continuity

Complete the following exercises, beginning on page 84:

1, 3, 11, 13, 15, 19, 23, 31

Routine Practice

02 - Continuity.pdf

Section 2.4: Rates of Change, Tangent Lines, and Sensitivity

Lesson Objectives

Calculate the average rate of change of a function.
Determine the slope of the tangent line at a point using limits.
Determine the equation of the tangent line to a curve at a given point.
Determine the equation of the normal line to a curve at a given point.

Section 2.4

Complete the following exercises, beginning on page 93:

1, 3, 7, 9, 13, 15, 19, 23, 25, 27, 29, 31, 33, 39

Routine Practice

03 - Average Rates of Change.pdf

Average Rates of Change

04 - Tangent Lines.pdf

Tangent Lines

04 - Normal Lines.pdf

Normal Lines

Quick Quiz: Sections 2.3 and 2.4

Unit 2 Test

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