Dave4Math / Courses / Limits and Continuity

Limits and Continuity

Master limits and continuity with this ULTIMATE introduction to the calculus. In this course, you'll learn different types of limits and about continuous functions.


January 9, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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You just found out your calculus teacher is exceedingly dull this semester. What are you going to do? Go. Introducing Dave’s online math course: Limits and Continuity. Dave is fun, you will enjoy him as a teacher, and he knows calculus really well. Now you can raise your expectations.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in limits and continuity.
  • Students who are enrolled in calculus 1 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 2.
  • Anyone wanting to learn techniques for finding limits and about continuous functions.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Limits and Continuity

  • Intuitive Definition of Limit
  • One-Sided Limits
  • Using Graphing Utilities to Evaluate Limits
  • Computing Limits Using the Laws of Limits
  • Limits of Polynomial and Rational Functions
  • Limits of Trigonometric Functions
  • Precise Definition of a Limit
  • A Geometric Interpretation
  • Continuous Functions
  • Continuity at a Number
  • Continuity at an Endpoint
  • Continuity on an Interval
  • Continuity of Composite Functions
  • Intermediate Value Theorem
  • An Intuitive Look at Tangent Lines and Rates of Change
  • Estimating the Rate of Change of a Function from Its Graph
  • More Examples Involving Rates of Change
  • Defining a Tangent Line
  • Tangent Lines, Secant Lines, and Rates of Change

Course Description

We begin this course with an intuitive introduction to limits. First, we study limits using a table of values and or graphs to understand what a limit is. Then we look at one-sided limits and two-sided limits and especially make use of piecewise-defined functions.

Next, we begin studying how to compute limits. We learn the limit laws and the limits of polynomial and rational functions. We also study finding limits using rationalization and other algebraic approaches. Limits of trigonometric functions and squeeze theorem are also examined.

After using tables of values and graphs to get an intuitive idea of the limit, we then clearly show how that approach is ambiguous. We motivate the precise definition of a limit using epilson-deltas. Several examples are illustrated by proving the value of a limit using the precise definition of a limit. We also show how a limit doesn’t exist by considering a piecewise function and applying the limit’s precise definition. We end this lesson by proving the limit laws (using the precise definition of a limit).

After a thorough understanding of limits we will begin to study continuous functions. First, we discuss continuity at a number, and then continuity at an endpoint, and also continuity on an interval. We next explain the continuity of the composition of functions, including the sum, product, and quotients. We also consider the continuity of polynomial and rational functions and the continuity of trigonometric functions. Towards the end of this lesson, we explore the Intermediate Value Theorem, and we end with a discussion on the existence of zeros of a continuous function.

We end this course with a discussion of tangent lines and rates of change. After that, we explore estimating the rate of change of a function from its graph and explaining the tangent line. Finally, we end this course with a detailed explanation of the difference between an average rate of change and an instantaneous rate of change by working through several examples.

Recommended Prerequisites for Limits and Continuity

I recommend the prerequisite course Functions and Their Graphs. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 1 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.