Lesson Video: Limits and Limit Notation Mathematics

Video Transcript

Limits and Limit Notation

In this video, we’ll practice using limit notation and explore the concept of a limit. Limits are very important tools which are used frequently in calculus. In many cases, they’re used as building blocks for more sophisticated concepts which will follow on from this video. The standard notation for a limit is as follows. We would read this mathematical statement as: the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. This statement is defined to be true, provided that we can make the value of 𝑓 of 𝑥 arbitrarily close to 𝐿 by taking 𝑥 sufficiently close to 𝑎, from both sides, without letting 𝑥 be equal to 𝑎.

Here, it’s worth mentioning that, in this video, we won’t be going through a precise mathematical definition of a limit. You might see this elsewhere using the Greek symbols 𝜀 and 𝛿. Instead, we’ll be working with this commonly used definition, which gives us a path to interpret the meaning of a limit. Consider the graph of some function 𝑓 of 𝑥. When our definition says that we can make the value of 𝑓 of 𝑥 arbitrarily close to 𝐿, this means as close to 𝐿 as we like. Loosely speaking, our limit is telling us that the value of 𝑓 of 𝑥 gets closer and closer to 𝐿 as the value of 𝑥 gets closer and closer to 𝑎. When we say from both sides, we mean that 𝑥 can either approach the value of 𝑎 from the positive direction, so when 𝑥 is greater than 𝑎, or from the negative direction, so when 𝑥 is less than 𝑎.

Finally, it’s very important that 𝑥 is not equal to 𝑎. And we’ll be returning to this fact later. Before moving on, note that you might see alternative notation for a limit, which would look like so. We would read this statement: 𝑓 of 𝑥 approaches 𝐿 as 𝑥 approaches 𝑎. Let’s now take a look at a quick example to illustrate the use of limit notation.

What is the correct notation that describes the following statement? As 𝑥 approaches zero, 𝑓 of 𝑥 approaches negative six.

For a question of this type, the first thing we might notice is this word approaches. Whenever we are told that the value of a function or a variable approaches something, it gives us a hint that our question might involve limits. The standard notation for a limit is shown here. And we would read the statement as: the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. Breaking it down, what this statement is telling us is that as the value of 𝑥 approaches the constant, here called 𝑎, the value of 𝑓 of 𝑥 will approach 𝐿. This 𝐿 is sometimes referred to as the value of the limit. A quick tip here is to remember that we’re not being told that 𝑓 of 𝑥 is equal to 𝐿, but rather that the value of our limit is equal to 𝐿.

Okay, now that we understand how to write a limit and how to interpret the statement, let’s see how it applies to our question. We are told that as 𝑥 approaches zero, 𝑓 of 𝑥 approaches negative six. This zero is the value which we consider 𝑥 to be approaching. And it’s represented by 𝑎 in our general limit equation. Similarly, this negative six, which is the value 𝑓 of 𝑥 approaches, is represented by 𝐿 in our general limit equation. We therefore write the following statement. The limit as 𝑥 approaches zero of 𝑓 of 𝑥 is equal to negative six. Since we have substituted 𝑎 is zero and 𝐿 is negative six, the statement is interpreted in the following way. As the value of 𝑥 approaches zero, the value of 𝑓 of 𝑥 approaches negative six.

Looking at the statement in the question and the statement that we have just written, these exactly match. This means that we have expressed the statement given in the question using the correct notation for a limit.

Let us now return to our definition to highlight a very important feature. Now, a crucial part of our definition is that the limit concerns values of 𝑥 which are arbitrarily close to 𝑎 but not where 𝑥 is equal to 𝑎. This means that the value of the limit, which of course is 𝐿, gives us useful information about the value of our function when 𝑥 is close to 𝑎. But we should not draw any conclusions about the value of our function when 𝑥 is equal to 𝑎. And, of course, this is 𝑓 of 𝑎. In fact, the value of 𝑓 of 𝑎 and the value of 𝐿 could be very different. And we can illustrate this with the following example.

Earlier, we saw a diagram which illustrated our limit definition in graph form. Let’s now assign some numbers to this graph to get more of a hand toward on the concept that we’re dealing with. We have that the limit has 𝑥 approaches one of our function, which we’ll now call 𝑓 one of 𝑥, is equal to three. Of course, this means that as the value of 𝑥 approaches one, the value of 𝑓 one of 𝑥 approaches three. Now, for this function, it just so happens that 𝑓 one of one is equal to three. Which means that when 𝑥 is actually equal to one, the value of our function is equal to three. However, this isn’t always the case. Let us consider some other function 𝑓 two of 𝑥.

𝑓 two of 𝑥 is almost identical to our first function, 𝑓 one of 𝑥. However, we notice that we have a hollow dot on our graph at the coordinate one, three. And instead, we have a filled dot on our graph at the coordinate one, four. What this means is that our function is not defined at the hollow dot but rather is defined at the filled dot. When 𝑥 equals one, our function equals four. So 𝑓 two of one equals four. Despite this difference, we should not be fooled into thinking that the value of our limit as 𝑥 approaches one is also equal to four. Actually, the fact that only 𝑓 of one has changed does not affect our limit at all. And we would still say that the limit as 𝑥 approaches one of 𝑓 two of 𝑥 is equal to three.

This is because the limit concerns values of 𝑥 which are arbitrarily close to 𝑥 equals one but not where 𝑥 is actually equal to one. As 𝑥 approaches one, 𝑓 two of 𝑥 still approaches three. We can extend this concept even further by noting that our function may not even be defined at a point where a limit is being taken.

Consider one final function 𝑓 three of 𝑥. This function is identical to our previous two functions, apart from when 𝑥 is equal to one. In this case, our function is undefined. So 𝑓 three of one is undefined. As with 𝑓 two of 𝑥, this change does not affect our limit. And we would still say that the limit as 𝑥 approaches one of 𝑓 three of 𝑥 is equal to three. Now, looking at these three examples, we see that we have three distinct cases. First, where the value of the function is equal to the value of the limit at that point. Next, where the value of the function is finite but not equal to the value of the limit at that point. And finally, where the value of the function is undefined at the point at which the limit is being taken.

We can generalize these three cases to say that if the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿, 𝑓 of 𝑎 could be equal to 𝐿, 𝑓 of 𝑎 could be finite but not equal to 𝐿, or 𝑓 of 𝑎 could be undefined. Hopefully, this example should illustrate why we should not draw any conclusions as to the value of 𝑓 of 𝑎 based on the value of our limit. Since there are many different possibilities. Let us now use this information to answer the following question.

True or false: If the limit as 𝑥 approaches five of 𝑓 of 𝑥 is equal to negative three, then 𝑓 of five must be equal to negative three.

For this question, we’re given information in the form of a limit. So let us interpret this statement. What this is telling us is that as the value of 𝑥 approaches five, the value of 𝑓 of 𝑥 approaches negative three. Let us now look at the second part of our question. The question is asking us if the limit guarantees that when 𝑥 equals five, the value of the function will be negative three. To answer this, let us recall the general form of a limit.

We recall that the limit concerns values of 𝑥 which are arbitrarily close to 𝑎 but not when 𝑥 is equal to 𝑎. In our question, this 𝑎 is represented by five. The limit gives us information about our function as 𝑥 approaches five but does not give us information about our function when 𝑥 is equal to five. In fact, 𝑓 of five could be equal to our limit, which is negative three. It could take any other value or it could even be undefined. Since we cannot guarantee that the value of 𝑓 of five is equal to negative three based on the limit, the answer to the question is false. 𝑓 of five does not have to be equal to negative three.

Now that we more fully understand the properties of limits, let us take a look at an example where we’re asked to evaluate a limit.

The following figure represents the graph of the function 𝑓 of 𝑥 is equal to 𝑥 squared. What does the graph suggest about the value of the limit as 𝑥 approaches two of 𝑓 of 𝑥.

For this question, we have been given a function. And we’ve been asked to evaluate a limit of said function. To interpret this, let us recall the general form of a limit. The limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. What this statement tells us is that the value of 𝑓 of 𝑥 will approach 𝐿 as the value of 𝑥 approaches 𝑎 from both sides. But remember, we’re not concerned with the point where 𝑥 is actually equal to 𝑎. Let us now apply this statement to our question.

The limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to some value. And we’ll call this value 𝐿 one. This 𝐿 one is the thing we need to find. What the statement is telling us is that the value of 𝑓 of 𝑥 approaches 𝐿 one as the value of 𝑥 approaches two from both sides. And remember, this doesn’t necessarily have to be the same as the value of our function when 𝑥 is equal to two. To find the value of 𝐿 one, we can see what happens to the value of our function as we get closer and closer to 𝑥 equals two. We’ll start by considering some value of 𝑥 which is slightly smaller than two.

Let’s say 𝑥 equals 1.5. In this case, 𝑓 of 𝑥 equals 2.25. Since we know that 𝑓 of 𝑥 equals 𝑥 squared, we could even verify our graphical findings by taking 1.5 squared. Let us now increase our value of 𝑥. So we’re approaching 𝑥 equals two from the left, so when 𝑥 is less than two. As we do this, we might begin to notice that the value of 𝑓 of 𝑥 appears to be approaching four. If we were to follow a similar process, starting with the value of 𝑥 which is slightly larger than two and then we were to approach 𝑥 equals two from the right. We would notice that the values of 𝑓 of 𝑥 also approach four.

Without really going into any calculations, we see that as 𝑥 gets closer and closer to two, 𝑓 of 𝑥 gets closer and closer to four. This is true if we approach from the left or from the right, so from both sides. Essentially, on our graph, we’re converging on the point two, four. Okay, so we have just shown that the value of 𝑓 of 𝑥 approaches four as the value of 𝑥 approaches two. Since this statement defines our limit, we can use our finding to say that the value of the limit is also four. In doing this, we have answered our question. Using our graph, we observed that the value of 𝑓 of 𝑥 approached four as the value of 𝑥 approached two from both the left and the right. We use this to say that the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to four.

Let us now take a look at one final example where we’re asked to evaluate a limit.

The following figure is the graph of the function 𝑓 where 𝑓 of 𝑥 is equal to sin 𝑥 over 𝑥. Part i, what is the value of 𝑓 of zero?

For part i of this question, we’re being asked to find the value of 𝑓 of zero. Which is to say, we must evaluate our function when 𝑥 is equal to zero. We can graphically find the value of 𝑓 of zero in the following way. We take the point on the 𝑥-axis where 𝑥 is equal to zero and draw a line up to meet our curve. When we do this, we notice that we reach a hollow dot.

What this means is that our function is not defined at this point on our curve. If we could see a solid dot on our graph at some other point where 𝑥 is equal to zero, that would mean our function would be defined here instead. However, we do not see a solid dot anywhere when 𝑥 is equal to zero. Which must mean that our function is undefined when 𝑥 is zero. Given this fact, we cannot assign a value to 𝑓 of zero, and we must simply say it is undefined. This is the answer to part i of our question. Let us now move on to part ii.

What does the graph suggest about the value of the limit as 𝑥 approaches zero of 𝑓 of 𝑥?

To better answer this part of the question, let us write out the general form of a limit equation. What this statement tells us is that the value of 𝑓 of 𝑥 will approach 𝐿 as the value of 𝑥 approaches 𝑎 from both sides. But we’re not concerned with the point where 𝑥 is equal to 𝑎. Okay, We want to find the limit as 𝑥 approaches zero of our function. In other words, the 𝑎 in the general form of our limit equation is zero. Okay, to find the value of our limit, we need to find 𝐿. Perhaps we’ll call this 𝐿 one to be clear. 𝐿 one is the value that 𝑓 of 𝑥 approaches as 𝑥 approaches zero from both sides. To find this 𝐿 one, we can look at our graph and see what happens to our curve as the value of 𝑥 approaches zero.

We see that as 𝑥 approaches zero, 𝑓 of 𝑥 approaches one. In other words, we’re getting closer and closer to the coordinate point zero, one. But wait, this point zero, one on our graph is a hollow dot, which means our function is not defined here. In actual fact, this does not cause us any problems. This is because our limit concerns values of 𝑥 which are arbitrarily close to zero but not where 𝑥 is actually equal to zero. Great, we have found that 𝑓 of 𝑥 approaches one as 𝑥 approaches zero from both sides. Which means, the value of 𝐿 one is equal to one. We can now rewrite our limit equation in full. The limit as 𝑥 approaches zero of 𝑓 of 𝑥 is equal to one. We have now answered both parts of our question.

We used a graph first to determine that 𝑓 zero is undefined. And then to conclude that the limit as 𝑥 approaches zero of 𝑓 of 𝑥 was equal to one. It should be worth noting here that the value of the limit as 𝑥 approaches zero of 𝑓 of 𝑥 was not equal to the value of the function when 𝑥 was equal to zero. We should always remember that the limit of a function as 𝑥 approaches some value, let’s say 𝑎, does not necessarily give us reliable information about the value of the function when 𝑥 is equal to 𝑎. Mistakenly concluding that these two things are always equal can sometimes lead us into trouble. In fact, we’ve seen in this question that our function does not even need to be defined at a point where a limit is being taken.

Okay, to finish off this video, let’s go through some key points. Limits are an important building block across many areas of calculus. The standard notation for a limit is shown here. And we would read this as: the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 is equal to 𝐿. You could also see the same information represented using slightly different notation. An interpretation of these statements is that the value of 𝑓 of 𝑥 approaches 𝐿 as the value of 𝑥 approaches 𝑎. And remember, this must be true as 𝑥 approaches 𝑎 from both sides, so from both the positive and the negative direction.

A crucial part of our definition is that the limit concerns values of 𝑥 which are arbitrarily close to 𝑎 but not where 𝑥 is equal to 𝑎. This means that the limit can give us useful information about our function for values of 𝑥 near 𝑎. But we should not draw conclusions about 𝑓 of 𝑎 itself, so the value of our function when 𝑥 is equal to 𝑎. We illustrated this earlier by showing that if the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥 was equal to 𝐿. It could be true that 𝑓 of 𝑎 was also equal to 𝐿, 𝑓 of 𝑎 was not equal to 𝐿 but equal to some other finite value, or, in fact, 𝑓 of 𝑎 was undefined.