Intuitive Introduction to Limits (The Behavior of a Function) [Video]

Using tables of values and graphs of functions, I explain what limits are and how to use them. Then I discuss one-sided limits, two-sided limits, and when limits do not exist. I use several illustrations to develop intuition with limits. Because these methods of estimating limits are not rigorous, we explain how estimating can lead to incorrect results.

Do you know how to study the behavior of quantities under a process of change? Do you know how to describe the behavior of a function on its domain? In this video, I’m going to give an intuitive introduction to limits, based on using tables of values and graphs.

Hi, I’m Dave, welcome back to my channel. In today’s episode, I’ll explain limits by estimating using tables and graphs. In this intuitive introduction to limits, you will better understand what limits are and how to find them. I also illustrate the Two-Sided Limit Theorem by working through several examples. Here I study one-sided limits and two-sided limits with emphasis on graphs. Then, I provide different cases when a limit may not exist. So this video is for anyone just starting a Calculus 1 course.

Let’s get started.

FAQ Intuitive Introduction to Limits

How do you introduce limits?

Most students usually want to understand limits before studying techniques concerned with finding limits. Using tables of values and graphs is the best way to begin understanding limits. This approach is an opportunity to gain intuition in terms of what a limit is and why they are essential.

What is the concept of a limit?

intuitive introduction to limits Limits Calculus Starts with Limits graph showing a limit

Limits are used to understand the behavior of a function on its domain. For example, if I make small changes in the input, we can ask how are the outputs changing? In this way, we can understand how a function behaves around an input value or long term behavior.

How do you explain limits in calculus?

The two focal points in calculus are the “slope of the tangent line” and the “area under a curve.” These are modern ideas that have a great deal of flexibility, in that they arise in many critical applications. Both these ideas are limiting processes, and in this way, we can explain the importance of limits in calculus.

What is the purpose of a limit?

Calculus is the study of change, and limits are the tool that calculus uses to frame the most critical problems. Limits allow us to mathematically model the study of real word quantities under a process of change. In applying calculus (using derivatives and integrals) to the real world, limits become the foundational idea.

Conclusion

We started without any understanding of what a limit is but knowing what functions are from precalculus. Using tables of values and graphs, we were able to get an intuitive introduction to limits. We were also able to see the shortcoming of these approaches and understand why a rigorous approach is needed.

We finished by understanding what one-sided limits and two-sided limits are. You also have a basic understanding of the cases when a limit might not exist, such as jumps, unbounded behavior, and oscillating behavior.

Next, you should be studying different techniques for finding limits. These techniques will be algebraic and trigonometric based and not based on using tables and graphs. Then, you will be ready for the critical concept of continuity. Keep working hard.

Now I wanna turn it over to you.

When intuitively trying to find a limit, which do you like better: tables or graphs? Either way let us know what you think in the comments below right now.

If you like this article or the video, please like or leave a comment and subscribe to my channel.

David A. Smith at Dave4Math

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.

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