Limits - Intro | Jake's Math Lessons

Limits are an important topic to understand in calculus. The reason for this is that most of the other main categories in calculus are built around limits.

But what is a limit?

When you take the limit of a function, we are essentially asking: what $y$ value does this function approach as $x$ gets really really close to a certain value.

Now we’ll get into a few examples of what this looks like graphically. If you’d like to learn more about evaluating limits algebraically, I’d recommend starting with the 8 limit properties.

Example 1

For example, take the function graphed below, $y = x^2$ . Imagine we travel along the function, from both sides, near $x=2$ . As we get closer and closer to the $x$ value of $2$ , what $y$ value do we also get closer and closer to?

We get closer and closer to a $y$ value of $4$ . This is a simple example, but this is the general idea behind all limits. If you need to find a limit of a function as $x$ approaches a certain value, you can figure this out quite easily with a graph of the function.

Just trace along the function from both the left and right side of that $x$ value and move in towards the specific $x$ value. Whatever $y$ value you close in on is the value of the limit of that function as $x$ approaches the given value.

In this example, we would say: “the limit of $x^2$ , as $x$ approaches $2$ , is $4$ .” Or “the limit as $x$ approaches $2$ , of $x^2$ , is $4$ .” In mathematical notation, that phrase would look something like this:

$$\lim_{x \to 2} x^2 = 4$$

It just so happens that the value of the limit is the same value we would get by plugging $x = 2$ into this function. What I mean by that is, $f(2) = 2^2 = 4$ is exactly what we got for the limit of $x^2$ as $x$ approaches $2$ . This is not a coincidence. I will go into more detail about why this is important when we discuss continuous functions and continuity. But for now, let’s do a few harder examples.

Example 2

I’m going to stick with a couple more examples of finding limits using graphs, because it is a skill you will need to know, and I’ve noticed that it’s something a lot of introductory calculus students have a hard time with. So if finding limits from graphs is something that confuses you, don’t worry.

Let’s try finding a little more challenging limit. Below is a graph of $f(x)$ .

Using the graph in Figure 1.2, we will consider two different limits. First let’s consider the following:

$$\lim_{x \to 1}f(x)$$

Let’s look at what’s going on in this graph at and around $x = 1$ . It looks like the function was supposed to go through the point $(1,\ 2)$ , but that point got taken out. Instead, there is a hole there, and the graph of $f(x)$ includes the point $(1,\ 4)$ . Therefore, we can say $f(1) = 4$ . All this means is that this function returns a value of $4$ , or $y = 4$ , when $x = 1$ . However, this has no impact at the limit we are considering.

Limits aren’t impacted by what happens at a specific point, only by what is happening around that point. As a result, knowing that $f(1)=4$ doesn’t help us to find

$$\lim_{x \to 1}f(x).$$

So what do we need to look at?

To find this limit, we need to look at what’s happening around $x = 1$ , and ignore what’s happening at $x = 1$ . As we get closer and closer to $x = 1$ , from the left and the right, what $y$ value do we get close to? Imagine traveling along this function, perhaps starting at $x = 2$ , and moving to the left. As you pass $x = 1.5$ , then $x = 1.25$ , then $x = 1.125$ , and so on, we get closer and closer to the hole I mentioned earlier. We get closer and closer to a $y$ value of $2$ .

The same thing happens if we start on that function at $x = 0$ and move to the right, toward $x = 1$ . From both sides we get closer and closer to $y = 2$ .

It is important to realize that we never actually get to $y = 2$ . This is because we never get to $x = 1$ , just infinitely close to it. As I said before, when solving for a limit, it doesn’t matter what happens at the point it’s asking about. It only matters what happens as you get infinitely close. This is the reason why, in this example, $f(1) = 4$ , but

$$\lim_{x \to 1}f(x) = 2.$$

The other important thing to point out in this example is that we approached $x = 1$ from both sides and they both led us closer and closer to the same $y$ value of $2$ . If both sides didn’t give us the same value, this limit would not exist. This is illustrated in greater detail in my lesson about one-sided limits.

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If you would like to continue reading about limits, check out my limits page. There’s plenty more lessons and example problems to read there. If your questions are not answered there, I’d love your feedback. Email me at jakesmathlessons@gmail.com with any unanswered questions you have and I’ll make sure to answer your question!

Limits – Intro