Computing Limits Algebraically

In section 10.1, we examined tables and graphs to help us evaluate limits. For instance, to evaluate the limit

we could construct a table of approximate y values near x = 0.

We could also graph the function to get an idea of how the y values behave as x gets closer and closer to 0.

For either representation, as x get closer and closer to 0, the corresponding y value gets closer and closer to 1. This means the value of the limit is 1,

If you look at the table or the graph, you’ll notice that the y value for the expression  is 1 at the value x = 0. This suggests that you could simply substitute  into the expression to evaluate the limit. By doing this, we can avoid creating the table or making the graph. In this section, we’ll examine the situations when this strategy works and ways to work around the problems that occur when it is not possible.

Read in Section 10.3

• How can a limit be computed algebraically?
• How do you evaluate limits involving difference quotients?
• Handout: Some Limits with Square Roots (PDF)
• Handout: More Complicated Difference Quotients – Polynomials (PDF)
• Handout: More Complicated Difference Quotients – Rational Function (PDF)
• Handout: More Complicated Difference Quotients – Square Root Functions (PDF)
• Handout: More Complicated Difference Quotients – Polynomials (PDF)

Section 10.3 Workbook (PDF) 8-18-19

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Basic Concepts of Limits

In this chapter, we’ll examine the concept of a limit. In its simplest interpretation a limit indicates how the output of a function behaves as the input is changed. Limits help us to spot patterns in this behavior.

We’ll begin this journey by using tables and graphs to relate the input and output of a function. These forms make it easy to spot patterns in the relationship. Unfortunately, these representations take a bit of work to set up. In later sections we’ll learn other methods that can help us to evaluate a limit.

Read in Section 10.2

• How do you evaluate a limit from a table?
• How do you evaluate a limit from a graph?
• Handout: Finding Limits Graphically (PDF)

Section 10.2 Workbook (PDF) 8-18-19

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Applications

Functions

Functions are used throughout calculus to model different quantities like cost, revenue, and profit. We need to review the basic concepts around functions before we can examine limits and other calculus concepts.

We start by examining the role of independent and dependent variables. Once we understand variables, we can use them to define functions.

Read in Section 10.1

• What is a function?
• What is function notation?
• Handout: Use Function Notation (PDF)
• What is a piecewise defined function?

Section 10.1 Workbook (PDF) 8-18-19

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Applications

Section 1 – Average Rate of Change

Section 2 – Instantaneous Rate of Change

Section 3 – The Derivative at a Point

Section 4 – The Derivative Function

Section 5 – Economic Applications of the Derivative

Section 6 – The Derivatives of Products and Quotients

Section 7 – The Chain Rule

Chapter 11 Workbook Practice Solutions (PDF) 8-23-19

Section 1 – Functions

Section 2 – Basic Concepts of Limits

Section 3 – Computing Limits Algebraically

Section 4 – Computing Limits at Infinity

Section 5 – Continuous Functions

Chapter 10 Practice Solutions (PDF) 8-18-19