This page explains how to thoroughly study basic limits and contains links to study material, videos and practice problems. You can use this page and the rest of this site to refresh your limits skills, learn limits on your own or supplement your course material.
Start with the study plan. This will take you through the topics in order and direct you to each page.
Important! Make sure you work through and understand all the practice problems on each page before going on to the next topic.
Limits Study Plan
Unit 1 - Getting Started
Work through the Pre-Limit Topics panel below.
Work through your textbook, referring to the Limits Core Material Panel below for comments on and links to these topics. If there are additional topics in your textbook not listed here, work through them as well.
Unit 2 - Overview Of Calculus
If there is a section in your textbook that gives you an overview of calculus, don't skip it. Having a big picture of where you are going can be very helpful.
Unit 3 - Definition of Limits
Read Topics 1-3 in the Limits Core Material Panel below.
Unit 4 - Finite Limits
Read the Finite Limits information under Topic 4 in the Limits Core Material Panel below.
Unit 5 - Continuity (including One-Sided Limits)
Read Topic 5 on One-Sided Limits and Topic 6 on Continuity in the Limits Core Material Panel below.
Unit 6 - Infinite Limits
Read the Infinite Limits information under Topic 4 in the Limits Core Material Panel below.
Unit 7 - Additional Topics
In the Limits Core Material Panel below, read the following topics.
Topic 7: Squeeze Theorem
Topic 8: What It Means When A Limit Does Not Exist
Topic 9: Removable and Nonremovable Discontinuities
Note: On the Discontinuities page there is a section discussing the similarity between zeros, holes and asymptotes. This discussion is not usually covered in traditional calculus courses since all this material is covered separately. However, to get it straight in your head, it helps to think about the relationship between the three, so that you can tell the difference easily. (Hint: I will usually have a question on the first exam related to this topic.)
Here are a few topics that you need to understand before working with limits.
1. Why Domain Is Important In Calculus
I don't know about you but when I was in algebra I kind of just blew off the subject of domain as not important and, therefore, not worthy of my time and energy (I know, not a good thought and can you believe a math teacher actually admitting this?!). However, in calculus, I found I couldn't ignore domain anymore. Consequently, I struggled to understand some topics because I didn't take the time to understand domain and it's role in calculus. So, if you are in that situation, do not despair. I have a section on the precalculus page to help get you up to speed (and it's not even that hard!). Even if you are comfortable with the concept of domain, it will really help you to go through this page as well.
Link To Why Domain Is Important
2. Piecewise-Defined Functions
Some people consider piecewise functions difficult and can't understand or graph them. However, they are not that hard. In the past, I have helped students understand them and, once they do, they look at me and say, is that it?! They can't believe how easy it is. So, if you are stumped by this topic, go to this page and see if this explanation helps. It is a good idea to go through this page to refresh your memory, even if you think you already understand this topic.
Link To Piecewise-Defined Functions
Limits Core Material
1. Formal Limit Definition
This is an important topic that some teachers may even skip (gasp!) as you are first learning limits. Do not allow yourself to be deprived of the joy of wrestling with this definition. It is not easy but studying and starting to understand it can expand your mind and prepare you for later calculus topics.
Link To Formal Limit Definition
2. Limit Notation
As you get further into calculus, you will find that notation is extremely important.
Link To Limit Notation
3. Limit Key
Here is a discussion of a key concept about limits which is important for every student to grasp.
Link To Limit Key
4. Finite And Infinite Limits
These pages discuss the two main types of limits. The terminology can sometimes be confusing and differ between textbooks and references. Here is what we mean when using these terms.
Finite Limits refer to limits where the variable is approaching a finite value. When the limit approaches positive or negative infinity, some textbooks refer to this as Infinite Limits.
Link To Finite Limits
Infinite Limits refer to limits where the variable is approaching infinity (or negative infinity). Some textbooks call these Limits At Infinity.
Link To Infinite Limits
5. One-Sided Limits
If you need to, go back and read the section on piecewise-defined functions. One-sided limits rely heavily on piecewise-defined functions. Once you think you understand them, read the one-sided limits page.
Link To One-Sided Limits
There are two topics on this page, Continuity and Discontinuities. You may skip the section on discontinuities if you are going through the study plan since it is covered in Topic 9 below.
Link To Continuity
7. Squeeze Theorem
Link To Squeeze Theorem
8. What It Means When A Limit Does Not Exist
There is a lot of confusing and contradictory information in textbooks, reference materials and websites about when a limit does not exist. Go to this page to help clarify it in your mind.
Link To Limits That Do Not Exist
9. Removable and Nonremovable Discontinuities
This section discusses the removable and nonremovable discontinuities using graphs. Also, you will find a discussion about the similarities between zeroes, holes, and vertical asymptotes in rational functions.
Link To Discontinuities