Unit 2: Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together.
A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct.
Completing this unit should take you approximately 19 hours.
Upon successful completion of this unit, you will be able to:
- determine the values of one- or two-sided limits for a function given by a graph;
- use algebraic methods to determine the values of one- and two-sided limits for a function given by a formula or state that the limit "does not exist";
- state whether a given function is continuous at a point, and use the properties of continuity to find limits and values of related functions;
- use the Intermediate Value Theorem to determine the number of times a function has a given value;
- approximate the roots of functions using the Bisection Algorithm;
- state the epsilon-delta definition of limit; and
- for a given epsilon, find the required delta graphically and algebraically for linear and quadratic functions.
2.1: Tangent Lines, Velocities, and Growth
Read this section on pages 1-7 for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4. For the solutions to these practice problems, see page 10-11.
2.1.1: Practice Problems
Work through the odd-numbered problems 1-9 on pages 7-9. Once you have completed the problem set, check your answers for the odd-numbered questions here.
2.1.2: Review
Before moving on, you should be comfortable with each of these topics:
- The Slope of a Tangent Line; pages 1-3.
- Average Velocity and Instantaneous Velocity; pages 3-5.
- Average Population Growth Rate and Instantaneous Population Growth Rate; pages 5-7.
- The Slope of a Tangent Line; pages 1-3.
2.2: The Limit of a Function
Read this section on pages 1-7 for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4. For solutions to these practice problems, see page 10.
2.2.1: Practice Problems
Work through the odd-numbered problems 1-19 on pages 7-9. Once you have completed the problem set, check your answers for the odd-numbered questions here.
2.2.2: Review
Before moving on, you should be comfortable with each of these topics:
- Informal Notion of a Limit; pages 1-3.
- Algebra Method for Evaluating Limits; pages 4-6.
- Table Method for Evaluating Limits; pages 4-6.
- Graph Method for Evaluating Limits; pages 4-6.
- One-Sided Limits; pages 6-7.
- Informal Notion of a Limit; pages 1-3.
2.3: Properties of Limits
Read this section on pages 1-8 to learn about the properties of limits. Work through practice problems 1-6. For the solutions to these problems, see page 14.
Watch this video on finding limits algebraically. Be warned that removing
from the numerator and denominator in Step 4 of this video is only legal inside this limit. The function 
is not defined at 
; however, when 
is not 4, it simplifies to 1. Because the limit as approaches 4 depends only on values of different from 4, inside that limit and 1 are interchangeable. Outside that limit, they are not! However, this kind of cancellation is a key technique for finding limits of algebraically complicated functions.Watch this video on limits as the slopes of tangent lines.
An earlier Khan Academy video (not used in this course) defined the limit that gives the slope of the tangent line to a curve as
at a point 
and called it the derivative of 
at 
. The text will introduce this term in Unit 3.
2.3.1: Practice Problems
Work through the odd-numbered problems 1-21 on pages 9-14. Once you have completed the problem set, check your answers for the odd-numbered questions here.
2.3.2: Review
Before moving on, you should be comfortable with each of these topics:
- Main Limit Theorem; page 1.
- Limits by Substitution; page 2.
- Limits of Combined or Composed Functions; pages 2-4.
- Tangent Lines as Limits; page 4 and page 5.
- Comparing the Limits of Functions; page 5 and page 6.
- Showing that a Limit Does Not Exist; pages 6-8.
- Main Limit Theorem; page 1.
Problem Set 2
Take this assessment to see how well you understood these concepts.
- This assessment does not count towards your grade. It is just for practice!
- You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.
2.4: Continuous Functions
Read this section on pages 1-11 for an introduction to what we mean when we say a function is continuous. Work through practice problems 1 and 2. For solutions to these problems, see page 16.
2.4.1: Practice Problems
Work through the odd-numbered problems 1-23 on pages 12-15. Once you have completed the problem set, check your answers for the odd-numbered questions here.
2.4.2: Review
Before moving on, you should be comfortable with each of these topics:
- Definition and Meaning of Continuous; page 1.
- Graphic Meaning of Continuity; pages 1-4.
- The Importance of Continuity; page 5.
- Combinations of Continuous Functions; pages 5-6.
- Which Functions Are Continuous?; pages 6-8.
- Intermediate Value Property; page 8 and page 9.
- Bisection Algorithm for Approximating Roots; pages 9-11.
- Definition and Meaning of Continuous; page 1.
2.5: Definition of a Limit
Read this section on pages 1-11 to learn how a limit is defined. Work through practice problems 1-6. For solutions to these problems, see pages 14-16.
Watch this video to learn the epsilon-delta definition of a limit.
2.5.1: Practice Problems
Work through the odd-numbered problems 1-23 on pages 12-14. Once you have completed the problem set, check your answers for the odd-numbered questions here.
2.5.2: Review
Before moving on, you should be comfortable with each of these topics:
- Intuitive Approach to Defining a Limit; pages 1-7.
- The Formal Definition of a Limit; pages 7-10.
- Two Limit Theorems; pages 10-11.
- Intuitive Approach to Defining a Limit; pages 1-7.
Problem Set 3
Take this assessment to see how well you understood these concepts.
- This assessment does not count towards your grade. It is just for practice!
- You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.