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Lesson Explainer: Limits and Limit Notation Mathematics • Higher Education

In this explainer, we will learn how to use limit notation and explore the concept of a limit.

Limits are one of the most fundamental tools in exploring the value of a function near an input value and are a building block of calculus. Before we formally define a limit, we can explore in the following diagram why limits might be useful.

This is a graph of the piecewise-defined function: 𝑓(π‘₯)=π‘₯+1,π‘₯<0,βˆ’π‘₯+1,π‘₯>0,2,π‘₯=0.

We can see from the graph (or the definition of the function) that 𝑓(0)=2. However, if we look at the diagram, we can see that if the inputs are near 0, then the outputs seem to have a different value. For example, 𝑓(0.1)=1.1 and 𝑓(0.01)=1.01.

We can see why this is the case from the graph; as the inputs get closer to 0 from the right, the outputs will get closer to 1. The same is also true as the inputs approach 0 from the left.

This means that if the input values are close to 0, we know the outputs will be close to 1. The idea of what happens to our function near (but not at) the input values is very important in determining information about its shape and is why we introduce the concept of a limit.

Definition: Limit of a Function

If the values of 𝑓(π‘₯) approach some value of 𝐿 as the values of π‘₯ approach π‘Ž (from both sides) but not necessarily when π‘₯=π‘Ž, then we say the limit of 𝑓(π‘₯) as π‘₯ approaches π‘Ž is equal to 𝐿 and we denote this as limο—β†’οŒΊπ‘“(π‘₯)=𝐿.

In the above example, we saw that as the values of π‘₯ approached 0, the outputs of the function approached 1, so we can say limο—β†’οŠ¦π‘“(π‘₯)=1. It is important to reiterate that the value of the function at 0 has no effect on the limit of the function as π‘₯ approaches 0, since we are only interested in the outputs near 0. So, in our case, even though 𝑓(0)=2, the limit of 𝑓 as π‘₯ approaches 0 is equal to 1.

Before we see how to apply this definition, we will discuss alternative common notation.

Definition: Alternative Limit of a Function Notation

The notation 𝑓(π‘₯)→𝐿 as π‘₯β†’π‘Ž has the same meaning as limο—β†’οŒΊπ‘“(π‘₯)=𝐿.

We can read the arrow as β€œapproaches” or β€œtends toward.”

Finally, there is another common definition of a limit, usually referred to as the πœ–βˆ’π›Ώ definition of a limit. This is a precise mathematical definition of a limit and, while helpful for proving limit results, it is less intuitive for working through examples and so we will not be covering this definition in this explainer.

Let’s start by seeing an example of how to write out the limit notation of a given function.

Example 1: Expressing Mathematical Statements Using Limit Notation

What is the correct notation that describes the following statement?

As π‘₯ approaches 0, 𝑓(π‘₯) approaches βˆ’6.

Answer

We recall that the limit of 𝑓(π‘₯) as π‘₯ approaches π‘Ž being equal to 𝐿 is denoted limο—β†’οŒΊπ‘“(π‘₯)=𝐿.

We are given that the values of π‘₯ are approaching 0 and the output values of 𝑓(π‘₯) are approaching βˆ’6, so we use π‘Ž=0 and 𝐿=βˆ’6, giving us limο—β†’οŠ¦π‘“(π‘₯)=βˆ’6.

It is also worth noting that we could also use the alternative notation 𝑓(π‘₯)β†’βˆ’6 as π‘₯β†’0.

In our next example, we will discuss what information we can determine about a function given the value of its limit.

Example 2: Understanding the Relationship between the Value of a Function and the Limit of a Function

If 𝑓(6)=βˆ’6, what can we say about limο—β†’οŠ¬π‘“(π‘₯)?

  1. limο—β†’οŠ¬π‘“(π‘₯)=βˆ’6
  2. limο—β†’οŠ¬π‘“(π‘₯)β‰ βˆ’6
  3. limο—β†’οŠ¬π‘“(π‘₯)=βˆ’1
  4. limο—β†’οŠ¬π‘“(π‘₯)=0
  5. We cannot draw any conclusions about limο—β†’οŠ¬π‘“(π‘₯).

Answer

We recall that limο—β†’οŠ¬π‘“(π‘₯)=βˆ’6 means that as the values of π‘₯ get arbitrarily close to 6 from either direction, the outputs of the function approach βˆ’6. This means that the limit of 𝑓(π‘₯) as π‘₯ approaches 6 tells us about the values of the outputs around 6 but not when π‘₯=6.

To clarify this, we consider the following curves. First, 𝑓(π‘₯)=βˆ’π‘₯.

In this diagram, as our values of π‘₯ approach 6 from either side, we can see that the outputs of our function approach βˆ’6.

In this case, limο—β†’οŠ¬π‘“(π‘₯)=βˆ’6 and 𝑓(6)=βˆ’6, the limit and the function agree, but this is not the only option. Consider the following graphs.

In this graph, we can see that 𝑓(6)=βˆ’6. In fact, we could represent this using a piecewise-defined function. However, if we evaluate the limit of this function as π‘₯ approaches 6, we notice it is not equal to βˆ’6.

As the inputs of the function approach 6 from either side, we can see the outputs approach βˆ’1. Hence, the graph indicates limο—β†’οŠ¬π‘“(π‘₯)=βˆ’1, which is not equal to the function at this point.

This means we cannot draw any conclusions about limο—β†’οŠ¬π‘“(π‘₯), which was option E.

Now, consider the statement limο—β†’οŠ¬π‘“(π‘₯)=βˆ’6. Can we deduce any information about 𝑓(6)? In this case, we consider the following graphs of functions.

In diagram (1), we can see that as the inputs approach 6 from either side, the outputs approach βˆ’6, so both limο—β†’οŠ¬π‘“(π‘₯) and 𝑓(6) are equal to βˆ’6.

However, in diagram (2), the outputs approach βˆ’6 as the values of π‘₯ approach 6, but we can see 𝑓(6)=4. So, in this case, the limο—β†’οŠ¬π‘“(π‘₯)=βˆ’6 and 𝑓(6)=4 are not equal.

Finally, in diagram (3), once again, as our inputs approach 6 from either side, the outputs approach βˆ’6, so limο—β†’οŠ¬π‘“(π‘₯)=βˆ’6. However, 6 is not in the domain of our function, so our function is not even defined at this value.

We can summarize this result in general as follows.

Property: Limit of a Function

If limο—β†’οŒΊπ‘“(π‘₯)=𝐿, then 𝑓(π‘Ž) does not have to be equal to 𝐿. In fact, 𝑓(π‘Ž) can be undefined.

In our next few examples, we will explore the relationship between the limit of a function and its graph.

Example 3: Understanding Limits in Relation to Graphs

The following figure represents the graph of the function 𝑓(π‘₯)=π‘₯.

What does the graph suggest about the value of limο—β†’οŠ¨π‘“(π‘₯)?

Answer

We recall that the notation limο—β†’οŠ¨π‘“(π‘₯)=𝐿 means that as the values of π‘₯ get arbitrarily close to 2, from either direction, the outputs of the function must approach 𝐿. The outputs of a function are represented by the 𝑦-coordinates of the curve, where the π‘₯-coordinate tells us the input of the function.

This means we can investigate the limit as π‘₯β†’2 of this function by considering the 𝑦-coordinates of the curve on either side of π‘₯=2. We will start with values of π‘₯>2. First, consider the value of the function when π‘₯=2.5.

From our diagram, we can see this is slightly above 6, and to evaluate the limit, we need the inputs to get arbitrarily close to 2. Let’s do this by trying values closer to 2.

We can see from the diagram that 𝑓(2.25)β‰ˆ5. We can continue in this manner.

As the input values approach 2 from the right, the output values approach 4. We can follow the same process for π‘₯<2.

We can see that 𝑓(1)=1 and 𝑓(1.5)=2.25.

As the input values approach 2 from the left, the outputs of the function approach 4.

Therefore, as the values of π‘₯ get arbitrarily close to 2, from either direction, the outputs of the function approach 4. Using limit notation, this is equivalent to the statement limο—β†’οŠ¨π‘“(π‘₯)=4.

Example 4: Understanding Limits in Relation to Graphs

The following figure is the graph of the function 𝑓, where 𝑓(π‘₯)=π‘₯π‘₯sin.

  1. What is the value of 𝑓(0)?
  2. What does the graph suggest about the value of limο—β†’οŠ¦π‘“(π‘₯)?

Answer

Part 1

We can determine the value of 𝑓(0) from the diagram. Recall that, for a graph of 𝑦=𝑓(π‘₯), any point on the curve has the coordinates (π‘₯,𝑓(π‘₯)), so we can determine the value of 𝑓(0) by inspecting the 𝑦-coordinate of a point on the curve with π‘₯-coordinate 0. We add the line π‘₯=0 to the diagram.

Since the curve has a hollow dot at the point (0,1), we know the function is not defined here. Therefore, there are no intersections between the line and curve.

Hence, 𝑓(0) is undefined.

Part 2

We recall that the notation limο—β†’οŠ¦π‘“(π‘₯)=𝐿 means that as the values of π‘₯ get arbitrarily close to 0, from either direction, the outputs of the function must approach 𝐿. We can determine the outputs of the function from its graph, so we can determine this limit by considering the graph on either side of π‘₯=0.

By considering the outputs of the function as π‘₯ approaches 0 from either side, as shown in the diagram below, we can see that the outputs approach 1.

It is important to reiterate that even though the function itself is not defined when π‘₯=0, this will not affect the limit since we are only interested in what happens when π‘₯ is arbitrarily close to 0 and not when π‘₯=0.

Hence, the graph indicates limο—β†’οŠ¦π‘“(π‘₯)=1.

Before we finish this explainer, it is worth exploring other options for evaluating limits. In the above examples, we used the graph of a function to determine its limit; however, to evaluate a limit, we only need to see what is happening to the outputs of the function. This means we can also investigate limits by taking sample points of the function.

For example, if we were not given the graph of 𝑓(π‘₯)=π‘₯π‘₯sin, where π‘₯ is measured in radians, we could investigate the limit of 𝑓(π‘₯) as π‘₯ approaches 0 by evaluating the function at values to the left and right of 0. First, 𝑓(0.2)=0.20.2β‰ˆ0.9933sin. We can then evaluate 𝑓 at a value even closer to 0 (e.g.,𝑓(0.1)=0.10.1β‰ˆ0.9983sin). We can continue this process to get the following table:

π‘₯0.20.10.01
𝑓(π‘₯)0.99330.99830.9999

We can see that as the input values approach 0 from the right, the outputs appear to approach 1. We can do the same for values below 0:

π‘₯βˆ’0.2βˆ’0.1βˆ’0.01
𝑓(π‘₯)0.99330.99830.9999

Once again, the outputs appear to approach 1. Hence, as both of these approach 1, the sample points seem to suggest that limsinο—β†’οŠ¦π‘₯π‘₯=1.

Let’s finish by recapping some of the important points of this explainer.

Key Points

  • If the values of 𝑓(π‘₯) approach some value of 𝐿 as the values of π‘₯ approach π‘Ž (from both sides) but not necessarily when π‘₯=π‘Ž, then we say the limit of 𝑓(π‘₯) as π‘₯ approaches π‘Ž is equal to 𝐿.
  • If the limit of 𝑓(π‘₯) as π‘₯ approaches π‘Ž is equal to 𝐿, we can denote this in two ways:
    • limο—β†’οŒΊπ‘“(π‘₯)=𝐿,
    • 𝑓(π‘₯)→𝐿 as π‘₯β†’π‘Ž.
  • The value of the function at π‘Ž does not affect its limit as π‘₯ approaches π‘Ž.
  • We can investigate the limit of a function at π‘₯=π‘Ž from its graph by considering what happens to the curve on either side of π‘₯=π‘Ž.
  • We can investigate the limit of a function by taking sample points above and below the value of the limit we are taking.

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