Limits

A limit describes the behavior of a function, or output, as the input gets closer and closer to a specific number, without necessarily reaching it. Limits let us analyze what happens at exact points of “instantaneous” change.

Definition

The limit of any function is defined as the value that the function approaches when the independent variable of the function approaches a particular value. A function’s limit exists only when the left-hand limit and right-hand limit of the function both exist and are equal.

Limits are used in calculus to define derivatives, continuity, and integrals, and they represent the value that a function approaches as the input approaches a certain value. Limits in math are of several types, each describing different situations and behaviors of functions as the independent variable approaches a certain value or infinity. Here are the main types of limits: one-sided limits, two-sided limits, infinite limits, and limits at infinity. [1]

The process of letting x or t approach a in an expression; the limit of a function 𝑓⁡(𝑥) as x approaches a is the value that 𝑓⁡(𝑥) approaches as x approaches a. [6]

We say that the limit of 𝑓(𝑥) is 𝐿 as 𝑥 approaches 𝑎, and write this as

$\lim_{x\,\rightarrow a} \,f\left( x \right) \,\,=\,\,L$

provided we can make 𝑓⁡(𝑥) as close to 𝐿 as we want for all 𝑥 sufficiently close to 𝑎, from both sides, without actually letting 𝑥 be 𝑎. [10]

Who

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]

Limit Instructional Videos

The YouTube Near-number instructional videos introduce the basic near-numbers: finite, infinite, and constant. These objects will allow you 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 (see the definition of the difference quotient below) 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.

$\frac{df}{dx}\,\,=\,\,\frac{\lim}{h\rightarrow 0}\,\,\frac{f\left( x+h \right) \,\,-\,\,f\left( x \right)}{h}$

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]

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 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.

$f\left( x \right) \,\,=\,\,\frac{x^2\,\,-\,\,1}{x\,\,-\,\,1}$

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

$\frac{\lim}{x\rightarrow 1}\,\,\frac{x^2-1}{x-1}=\frac{\lim}{x\rightarrow 1}\,\,\frac{\left( x+1 \right) \,\,\left( x-1 \right)}{x-1}=\,\,\frac{\lim}{x\rightarrow 1}\,\,\left( x+1 \right) =2$

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]

What is the purpose of learning limits before derivatives and integrals? Can these concepts be derived from each other and is there a practical application for this order of learning in mathematics?

The first ideas of a “limit” consist of just looking at an expression such as 2x + h then they consider what this expression becomes as the variable quantity “h” approaches zero.

In fact, a teacher can get a long way into basic calculus without formally writing the above at all!

I would like readers to watch my introduction to calculus lesson for 16-year-olds.

INTRODUCTION TO DIFFERENTIATION

Teaching mathematics this way is necessary if we expect to make Calculus more accessible to most students.

The main skill that we teach in basic calculus is the method of differentiating powers of x.

Since Integration is anti-differentiation, it seems logical to teach differentiation before integration. [9]

Why

See Theoretical Knowledge Vs Practical Application.

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.”

Algebra of Limit

Law of Addition

$\lim_{x\rightarrow a} \{f(x)+g(x)\}=\lim_{x\rightarrow a} f(x)+\lim_{x\rightarrow a} g(x)$

Law of Subtraction

$\lim_{x\rightarrow a} \{f(x)-g(x)\}=\lim_{x\rightarrow a} f(x)-\lim_{x\rightarrow a} g(x)$

Law of Multiplication

$\lim_{x\rightarrow a} \{f(x)\cdot g(x)\}=\lim_{x\rightarrow a} f(x)\cdot \lim_{x\rightarrow a} g(x)$

Law of Division

$\lim_{x\rightarrow a} \{f(x)\div g(x)\}=\lim_{x\rightarrow a} f(x)\div \lim_{x\rightarrow a} g(x)$

References

[1] [ ℰ ] “Limits in Calculus.” GeeksforGeeks, December 14, 2020. https://www.geeksforgeeks.org/maths/limits/.

[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] Strang, Gilbert, Joseph Lakey, Elaine A. Terry, et al. “Calculus Volume 1.” openstax, March 10, 2016. https://openstax.org/details/books/calculus-volume-1.

[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.

[9] Lloyd, Philip. 2024. “What is the purpose of learning limits before derivatives and integrals? Can these concepts be derived from each other and is there a practical application for this order of learning in mathematics?” Quora. Accessed July 13. https://qr.ae/ps0oQR.

[10] Dawkins, Paul. “Calculus I.” Section 2.2 : The Limit. Paul’s Online Notes, November 16, 2022. https://tutorial.math.lamar.edu/classes/calci/thelimit.aspx.

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.

[ ℰ ] “L’Hopital Rule in Calculus.” GeeksforGeeks, August 28, 2023. https://www.geeksforgeeks.org/maths/l-hopital-rule/.

[ ℰ ] “Limit Formulas.” GeeksforGeeks, February 22, 2022. https://www.geeksforgeeks.org/maths/limit-formula/.

[ ℰ ] “Limits by Direct Substitution.” GeeksforGeeks, June 2, 2021. https://www.geeksforgeeks.org/maths/limits-by-direct-substitution/.

“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.

[ ℰ ] Math in History. “A Simple Real World Example of Limit.” Medium, June 29, 2026. https://medium.com/@ganeshonline6/a-simple-real-world-example-of-limit-de9e64feb467.

Before we dive into the mathematics of calculus, let us set the stage with a hypothetical scenario. Imagine you are standing on the top floor of a 100-foot-tall house, holding a baseball signed by the legendary Babe Ruth. For historical context, Babe Ruth is one of the most famous baseball players in history, and memorabilia associated with him is incredibly valuable. A pristine baseball signed by him sold for over $300,000 at a Sotheby’s auction in 2012. The logical choice would be to secure the ball and sell it. However, for the sake of science, you decide to drop it out the window to measure how fast it falls. This scenario perfectly illustrates a problem that stumped mathematicians for centuries: how do we calculate the exact speed of a moving object at a single, frozen instant in time? The solution required a completely new branch of mathematics — calculus — and its foundational concept: the limit.

[ ℰ ] “Properties of Limits.” GeeksforGeeks, April 28, 2021. https://www.geeksforgeeks.org/maths/properties-of-limits/.

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

[ ℰ ] “Sandwich Theorem.” GeeksforGeeks, December 14, 2020. https://www.geeksforgeeks.org/maths/sandwich-theorem/.

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

Videos

Limits intro

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.

Limits, L’Hôpital’s rule, and epsilon delta definitions | Chapter 7, Essence of calculus

What is Calculus – Lesson 2 | Limits

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.

[ ℰ ] This exceptional reference is highly recommended for your consideration.