## Definition

Back when I was taking my introductory calculus courses, one of the first topics that we covered was a section on limits and rates of change.  To a math student, “rate of change” kind of makes some sense.  It sounds like a math concept that explains how fast something is changing.  It’s a fairly intuitive concept, and we will eventually see that it is a very important concept as well!  On the other hand, at first glance, the concept of a “limit” sounds a bit more abstract.  We all know about limits in terms of, for example, the speed limit for a car or other defined measures.  But how does this construct fit into calculus, or does “limit” mean something else entirely in this context?

The concept of a limit is essentially the same as when we’re talking about a speed limit.  Let’s say there is a posted speed limit of 50 mph.  Your car is not allowed to travel any faster than this.  So, imagine then what a graph would look like of the speed of your car over time.  At time = 0, you start stationary.  Then you accelerate and start driving a bit faster, until you get to the speed limit, at which point you don’t go any faster (because we obey the laws!), and then you approach a stop sign and slow down to come to a stop.  When you look at the graph of what has just happened, you can see that the speed limit is basically a speed value that your car approaches.1

## Who

Limits are needed to define continuity of a function, a function’s derivative, the definite integral (this can actually be done better without limits via Darboux integration, but this is seldom used in calc courses), and sequences/series/power series—i.e., the main elements of the rest of Calc I & II.

IMHO, the concepts surrounding limits are woefully undertreated in the calculus curriculum, which (along with students misunderstanding functions or making algebra errors, which only involve prerequisite material) is the main cause of a great deal of student confusion in calculus. The videos below help to fill in those gaps, providing a firmer foundation for all of calculus:2

Near-number instructional videos – YouTube

The Near-Numbers instructional video series introduces the basic near-numbers: finite, infinite, and constant. These objects will allow us to much more precisely discuss and explore the core concepts of calculus.

Limits are one of the core concepts of calculus as they can be used to derive the slope at a point, which is one of the key subjects of calculus. They can also be used to discern derivatives, which in turn are used throughout calculus.3

Limits allow us to study a number from afar. That is, we can study the points around it so we can better understand the given value we want to know.

Especially in derivatives, where change in position is purely relative, the points around a given value are critically important.

For an example, I want to know the slope of the point (0,0) on the function y=sin(x). By simply looking at that point, and nothing around it, it’s impossible to tell the slope. However, if we look at the points directly to the left and right of (0,0), we will see (with fairly rigorous computations) that the slope is in fact, 1.4

Limits are used to examine function behavior around points. But specifically, one limit (definition) is fundamental to Calculus: The Difference Quotient.

This is the definition of a derivative. Without limits, it would be very difficult to truly talk about rate of change. Therefore I say unto you, Calculus is impossible to do without knowing how to do limits.5

The Limit, is often learned in Calculus. The Limit is extremely important in Calculus since it is literally in Derivatives, L’hopitals Rule, Series and more!

I am a 10th grader in High School, taking BC Calculus this semester. The limit shows up everywhere! For example, when we were doing convergence and divergence of a series, the LIMIT comes up for many, many and many of our tests. For example, the nth term, integral test, alternating series, etc. All the tests must have a limit that has n approaching infinity.

Moving back, when we had our improper integral unit, we had to use the good ol limit!

For example, when we had a integral from 1 to infinity, we had to rewrite that integral as the limit of a letter, B, to that upper bound. Limits are what helps us solve the improper integral!

Moving back a little more, the limit is a significant part of AB calculus, as this is the 1st thing one learns. We have limits and their special cases, and what to do if an indeterminate form pops up.

Finally, in AB Calc, we are exposed to the formal and alternate form of the derivative, which is a limit, where h goes to 0.

In essence, LIMITS are a key leg of the calculus house, and without it, calculus isn’t calculus!6

## What

Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns about the behaviour of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas in indefinite the integrals are expressed without limits, and it will have an arbitrary constant while integrating the function. In this article, we are going to discuss the definition and representation of limits, with properties and examples in detail.7

What is the purpose of a limit in calculus?

A limit allows us to examine the tendency of a function around a given point even when the function is not defined at the point. Let us look at the function below.

Since its denominator is zero when x=1f(1) is undefined; however, its limit at x=1 exists and indicates that the function value approaches 2 there.

This tool is very useful in calculus when the slope of a tangent line is approximated by the slopes of secant lines with nearing intersection points, which motivates the definition of the derivative.8

## How

Many of the References and Additional Reading websites and Videos will assist you with how to evaluate limits.

As some professors say: “It is intuitively obvious to even the most casual observer.

## References

1 “What Are Limits In Calculus?” 2012. sk19math.blogspot.com. https://sk19math.blogspot.com/2012/05/what-are-limits-in-calculus.html.

2 Swenton, Frank. “What Is The Importance Of Limits In Calculus?” 2021. Quora. https://www.quora.com/What-is-the-importance-of-limits-in-calculus.

3 Esrig, Eli. “What Is The Importance Of Limits In Calculus?” 2021. Quora. https://www.quora.com/What-is-the-importance-of-limits-in-calculus.

4 Velcheck, Ryan. “What Is The Importance Of Limits In Calculus?” 2021. Quora. https://www.quora.com/What-is-the-importance-of-limits-in-calculus.

5 Coggins, Anthony. “What Is The Importance Of Limits In Calculus?” 2021. Quora. https://www.quora.com/What-is-the-importance-of-limits-in-calculus.

6 Hui, Anthony. “What Is The Importance Of Limits In Calculus?” 2021. Quora. https://www.quora.com/What-is-the-importance-of-limits-in-calculus.

7 “Limits In Calculus (Definition, Properties And Examples)”. 2021. BYJUS. https://byjus.com/maths/limits/.

8 “What Is The Purpose Of A Limit In Calculus? | Socratic”. 2021. socratic.org. https://socratic.org/questions/what-is-the-purpose-of-a-limit-in-calculus.

## Additional Reading

“An Intuitive Introduction To Limits – BetterExplained”. 2021. betterexplained.com. https://betterexplained.com/articles/an-intuitive-introduction-to-limits/.

“Calculus I – The Definition Of The Limit”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx.

“Calculus I – Limits “. 2021. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/Classes/CalcI/limitsIntro.aspx.

“Introduction To Limits – Calculus | Socratic”. 2022. socratic.org. https://socratic.org/calculus/limits/introduction-to-limits.

“Introduction To Limits In Calculus”. 2021. analyzemath.com. https://www.analyzemath.com/calculus/limits/introduction.html.

“Limits (An Introduction)”. 2021. Math Is Fun. http://www.mathsisfun.com/calculus/limits.html.

“List Of Common Limits”. 2022. planetmath.org. https://www.planetmath.org/ListOfCommonLimits.

Ryan, Mark. Calculus for Dummies. Hoboken, NJ: Wiley Publishing, Inc., 2003.

“SOME IMPORTANT LIMITS – Math Formulas – Mathematics Formulas – Basic Math Formulas”. 2022. pioneermathematics.com. https://www.pioneermathematics.com/some-important-limits-formula.html.

Strogatz, Steven. Infinite powers: how calculus reveals the secrets of the universe. Boston: Houghton Mifflin Harcourt: 2019.

## Videos

A limit tells us the value that a function approaches as that function’s inputs get closer and closer to some number. The idea of a limit is the basis of all calculus.

“Limits Intro (Video) | Limits And Continuity | Khan Academy”. 2021. Khan Academy. https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-2/v/introduction-to-limits-hd.

In Calculus, it’s extremely important to understand the concept of Limits. With an interesting example, or a paradox we could say, this video explains how Limits help us understand values that cannot be determined using basic math. Watch this video to get an understanding of the concept of Limits.