So, in the previous article, you were introduced to limits as the point to which a measure converges to as it approaches a certain value. What does that mean?

Limits

Let's look at the following image:

We're able to see that as x approaches the value 3, the function: f(x) or y, approaches -3. In other words, f(x) converges to -3 as x approaches 3. We're not saying that f(3) = -3, but rather saying that moving along the function towards x = 3, we are moving towards y = -3. Let's look at an example to show why this is important.

Limits at Holes (Discontinuity)

Here, we're able to see that f(-3.5) is undefined, but the limit still exists. Why? Well, let's look at our definition for a limit: As x approaches -3.5, f(x) approaches 2. ✅
So, even though f(-3.5) is undefined, because the function still converges to the point y = 2, the limit is defined as y = 2.

Basic Limit Notation

You know how f(x) denotes functions and √ symbolizes radicals? Similarly, limits have a special way of being written as well:

Now, this may seem intimidating, so let's break it down. The "lim" symbolizes that this is in fact, a limit. The x→c is read as "x approaches c" and indicates the value that x is converging toward. Finally, f(x) is the function that we intend to find the limit of. Let's read this expression mathematically: The limit as x approaches c of f(x). That's it! The intimidating syntax will soon be a simple phrase that you'll say from the tip of your tongue (whether you'd like to or not 😁). Now, let's move on to checking limits from both sides.

Breaking Down Limits

So you've understood that for a limit to be defined, it has to converge towards a point when x = some value. It doesn't matter if that point is defined on the function, but the values leading up (and down) to it must. If we want to break this definition of converging down, how could we do it?

Well, for convergence, we can break the function down into two parts: the left side, leading x up to the point, and the right side, leading x down to the point. Think of it like a marble ball being rolled across a track - if we roll from the left or right of the function, at the point we're interested in, they must be at the same height.

We can see the "purple marble" being pushed from each side of the function, rolling towards the same point at x = 2. To generalize this for all limits, we can state that there are two limits of a function: one from the left and one from the right. For a limit to be defined at x = c, the limit as x approaches c from the left must be the same as the limit as x approaches c from the right.

Here's a graph to illustrate that there can be a limit from the left, and a limit from the right, that are not the same.

If we look at the limit of the above function as x approaches -5 from the left, we can see it is a negative value, let's say: -2. However, the limit of the above function as x approaches -5 from the right is a positive value, approximately 2. Now, there must be a way to denote if you're moving from the left or right, right? Yep. Here's how:

Limit as x approaches -5 from the right:

Limit as x approaches -5 from the left:

Make sure you don't mix up the positive/negative signs; x from the right is positive, x from the left is negative.

You can see that the two limits are different, which means that the general limit as x approaches -5 DNE (does not exist). In other words:

TL;DR: the limit at a point can only exist if the limit from the right and the left are the same. 

Now, hopefully you've learned limit notation, how to identify the right and left limits, and to check if a limit exists. Next up, we'll cover continuities and a few theorems to aid us in finding limits. Additionally, be sure to check our our socials down below (we have insta and tiktok) for study tips, time management, and more! Join our Discord to get problems, answer questions, and network with a growing community!