Taking the Limit to Infinity!

Going through this topic, I did not use my own intellect to come up with these answers.  And trust me, people can spend infinitely many (pun intended) hours on rearranging and changing my wording.  It all depends on how you as a student or someone interested in math sees it.

If we want to take a limit of a function at a point, the limit exists if the tolerance level is not broken.  If we want to take the limit of the function (meaning x approaches positive or negative infinity) and the limit exists, then the function converges to that limit.  If we want to take the limit of the function but the limit does not exist, then there is no bound on the function so that it diverges off to positive or negative infinity (but this is not the same positive or negative infinity as the input values).

Let's see if someone else can sum up my ideas better:

Limits Research Project

This youtube video was the inspiration for my mathematical blog.  Our project is designed to explain a mathematical topic more in depth so that high school students can understand it and I thought a video was a great idea.  But then I

Limits(mathematics) Wikipedia is always a great place to start, but be careful friends because it can be deceptive sometimes!

Limits to Infinity Want some practice and more explanations? An online resource to limits of functions posted by Duane Kouba, a professor at UC Davis.

Limits in the Real World This is a youtube video of a project done for the same class two years previous to my class.

Definitions and Properties of Limits This is an online resources that compiles different definitions and properties of limits.

Introduction to Limits Once again, another project for the same course, just a few years before my time.

Other sources listed in the topic posts:

Limits Research Project

Youtube video of "Mean Girls"

Karl Weierstrass

Alright, time now for the big kahuna...trying to combine these two crazy ideas into one.  To help me do so, I first asked a few people to take a minute to describe what they meant together.  Here a few responses:

Me: How are limits and infinity related?

Answer 1: Well, we look at the limit as x or our input approaches infinity to see if the function (or sequence) is bounded.

Answer 2: If the function has no limit, like it diverges, then it goes of to positive or negative infinity

Answer 3: To understand the function fully, we typically look at end behavior to see what the function is doing as x gets larger and larger (infinity) or smaller and smaller (negative infinity).  That is taking the limit of the function as x approaches the respective end(s) of the input values.

Me: How are limits and infinity NOT related?

Answer 1: Sometimes we don't care about what's going on at infinity or negative infinity.  We want to take the limit as x approaches a particular REAL value in the function, not a superficial theoretical number.

Answer 2: Taking a limit of a function deals with infinity but taking THE limit AT a value that x approaches is a whole different ball game.

Answer 3: Infinity is just a theoretical convention we use to understand what is happening in the overall scheme.  The limit of something could occur at any point before positive or negative infinity.

So we can start to piece together what infinity has (and has not) to do with limits.  It really all depends on how you are talking about a limit.  If you want to talk about the limit of a function at a particular value, we need to remember Weierstrass and Cauchy and their tolerance of the function's outputs compared with the inputs.  If we want to think about the limit of the function, we need to consider as our inputs approach positive or negative infinity (end-behavior).  This will tell us if the function converges (whatever the value of the limit as x approaches(negative) infinity of f(x)) or diverges (goes to positive or negative infinity...).

Before reading, make sure you read the first part Topic 1: What does infinity mean to you?-An Intro

If you watched the video of Lindsey Lohan's ...acting... (posted below), you would see some crazy formula that involved a limit and the answer saying that the limit does not exist.  How can we discuss limits when they sometimes don't exist?  Doesn't that do against our nature of reasoning?  Well, let's discuss what a limit truly is.  Before that, however, here is an interview with a college graduate (named grad for privacy reasons) about limits and what they mean to her:

Me: What is a limit?

Grad: It's like a cap or a bound of something.

Me: Explain what you mean by a cap or a bound.

Grad: When I say a bound, I mean it is something that a graph or a function approaches. 

Me: What do you mean when you say it approaches it?

Grad: I mean that the function at a particular point that you "tell" the limit to approach is the value of the function.

Once again, just like our infinity exploration, finding a concrete definition of a limit is challenging. There are thousands of definitions out in the world so let's see if we can crack a few of them:

Wikipedia: The epsilon-delta definition by Karl Weierstrass states

Let ƒ be a function. To say that

means that ƒ(x) can be made as close as desired to L by making the independent variable x close enough, but not equal, to the value c.

How close is "close enough to c" depends on how close any one person wants to make ƒ(x) to L. It also of course depends on what function ƒ is and on what the value of c is. The positive number ε (epsilon) is how close one wants to make ƒ(x) to L; one wants the distance to be less than ε. The positive number δ is how close one will make x to c; if the distance from x to c is less than δ (but not zero), then the distance from ƒ(x) to L will be less than ε. Thus δ depends on ε. The limit statement means that no matter how small ε is made, δ can be made small enough.

The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work.  In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point.

This definition also works for functions with more than one input value. In those cases, δ can be understood as the radius of a circle or sphere or higher-dimensional analogy, in the domain of the function and centered at the point where the existence of a limit is being proven, for which every point inside produces a function value less than ε from the value of the function at the limit point.

Breaking this down, we can try to see what each part of the definition means to actually define a limit.  If you have a function f(x) and for whatever x you “pick” as it approaches some value c, which is the point at which you are trying to find the limit at, there is some small number greater than zero represented by delta.  This means that the if the difference of the chosen x and c remains smaller than that delta, the limit only exists if the value of the function and the value of the limit at c also remain in some small “tolerance” we choose to be epsilon. 

Breaking this down further, consider having two points you choose related to “input” values in which we can show their extreme closeness to each other (or the difference between these two points is some extremely small number).  For the limit to exist, the distance from the value of these two points when plugged into the function must also remain small (or f(x)-L, where L is the limit at c is some extremely small number as well).   If not, there will be a disjoint in the function.  In complete layman’s terms, “If you start close (the input), you have to end close (the output).”

The first topic(s) I want to discuss is limits and infinity.  I have asked several people what exactly infinity means to them and how can they describe it.  One interview is posted below:

(Below is a interview conducted with an Algebra II student)

Me: What does infinity mean?

Student: big.

Me: Can you describe it more in detail?

Student: No because there aren't enough words to describe infinity because it is huge.  Hence why it is called infinity.

Me: Do you think people struggle with the concept of infinity?

Student: Yes because like I said, we don't really have a good way of describing it.  In math, we kinda just pretend it is like a number and leave it at that.

Me: Is infinity a number?

Student: Yes, I mean no not really.  I mean it is a number but it's bigger than any number we can describe because it is infinity.

Me: Can you have an infinity that is larger than another infinity?

Student: Yes you can.

Me: How is that so if you are saying that infinity is a number?

Student: I guess that in theory infinity is some number just hanging around out there in space that does exist but in real life it doesn't exist.

Me: What do you mean it doesn't exist in real life?

Student: We can't actually describe it expect for the symbol of infinity.  So we know it is there but we can't write it like the number 2.  We know 2 is less than 3 but greater than 1.  1 infinity can be greater than another but we can't describe it using our numbers.

As you can see, describing infinity can be quite the daunting task.  How can we begin to explain infinity to a high schooler?  Maybe the answer lies in limits...

This blog began as a research project I began doing in my senior year at Virginia Tech. I am a math education major graduating in May 2012 and developed this blog to break down tough mathematical topics and create answers based on explanations high schoolers could understand.  Please feel free to interact with this blog and post questions, answers, and ideas. The following video almost was an inspiration for my blog.

Limits Research Project