### Video Transcript

Limits from Tables and Graphs

In this video, we will learn how to
evaluate the limit of a function using tables and graphs. We’ll be looking at a variety of
examples of how we can use tables and graphs in order to evaluate limits of
different functions. Using tables and graphs can be a
very nice visual way to find a limit. But before we move on to the use of
tables and graphs, let’s recall what a limit is.

This here is a limit. We can say that it is the limit as
𝑥 approaches 𝑎 of 𝑓 of 𝑥. And when we’re considering this
limit, what we’re actually thinking is the value 𝑓 of 𝑥 approaches as 𝑥 gets
closer and closer to 𝑎. Let’s start by discussing how we
can use a table to find a limit such as this one. We will do this by considering the
following example.

Estimate the limit as 𝑥 tends to
negative two of 𝑓 of 𝑥 from the given table.

As we can see from the table, we’ve
been given 𝑥-values getting closer and closer to negative two from above and from
below. And we’ve been given their
corresponding 𝑓 of 𝑥 values. Let’s start by considering the
𝑥-values from below negative two. We have that 𝑓 of negative 2.1 is
equal to 36.9. 𝑓 of negative 2.01 is equal to
36.09. 𝑓 of negative 2.001 is equal to
36.009. Here, our 𝑥-values are getting
closer and closer to two. We need to consider what’s
happening to our 𝑓 of 𝑥 values. We can quite clearly see that 𝑓 of
𝑥 is getting closer and closer to 36. Let’s now consider the 𝑥-values
just above negative two. We have that 𝑓 of negative 1.9 is
equal to 35.1. 𝑓 of negative 1.99 is equal to
35.91. 𝑓 of negative 1.999 is equal to
35.991. So again, here we can see that our
𝑥-values are getting closer and closer to negative two.

As this is happening to 𝑥, we need
to consider what’s happening to 𝑓 of 𝑥. Our 𝑓 of 𝑥 values go 35.1, 35.91,
and 35.991. Therefore, it’s safe to say that
these 𝑓 of 𝑥 values are tending towards 36. As we can see, as 𝑥 approaches
negative two from both directions, the value which 𝑓 of 𝑥 approaches agrees with
one another. They’re both 36. Therefore, we can say that as 𝑥
tends towards negative two, 𝑓 of 𝑥 approaches 36. And if we convert this into
mathematical notation, we arrive at our estimate. And that is that the limit as 𝑥
approaches negative two of 𝑓 of 𝑥 is equal to 36.

Let’s now consider another example
where we have to find the limit from a table, but with a slight difference.

Determine the limit as 𝑥 tends to
five of 𝑥 squared plus three 𝑥 over the square root of 𝑥 minus one by evaluating
the function at 𝑥 is equal to 4.9, 4.95, 4.99, 4.995, 4.999, 5.001, 5.005, 5.01,
5.05, and 5.1, rounding to the nearest three decimal places.

Now, our first step in answering
this question is to evaluate the function at the given 𝑥-values. Now, the function we need to
consider is the function inside the limit. And we can call this function 𝑓 of
𝑥. Therefore, 𝑓 of 𝑥 is equal to 𝑥
squared plus three 𝑥 over the square root of 𝑥 minus one. Let’s now draw a table of values
for the 𝑥-values given in the question along with their corresponding 𝑓 of 𝑥
values. And we can find these corresponding
𝑓 of 𝑥 values by simply plugging their 𝑥 values into 𝑓 of 𝑥.

Substituting 𝑥 is equal to 4.9
into 𝑥 squared plus three 𝑥 over the square root of 𝑥 minus one, then we obtain
that 𝑓 of 𝑥 is equal to 19.602. Substituting in 4.95, we obtain
that 𝑓 of 𝑥 is equal to 19.800. We mustn’t forget to round our
values of 𝑓 of 𝑥 to three decimal places, since this is what the question has told
us to do. Continuing this on, we find that 𝑓
of 4.99 is 19.960. 𝑓 of 4.995 is 19.980. 𝑓 of 4.999 is 19.996. Continuing this on for the five
remaining values, we get that 𝑓 of 5.001 is 20.004. 𝑓 of 5.005 is 20.020. 𝑓 of 5.01 is 20.040. 𝑓 of 5.05 is 20.200. And 𝑓 of 5.1 is 20.402.

Now, we’re evaluating the limit of
𝑓 of 𝑥 as 𝑥 approaches five. The value of five comes between the
values of 4.999 and 5.001. Let’s now look at the trend in the
𝑓 of 𝑥 values as 𝑥 gets closer and closer to five. As 𝑥 approaches five from below,
we can see that the 𝑓 of 𝑥 values are getting closer and closer to 20. And if we look at the 𝑓 of 𝑥
values as 𝑥 approaches five from above, then we can see that these 𝑓 of 𝑥 values
are also approaching 20.

We know that they are approaching
20 since with each step that we’re getting closer to five with our 𝑥-values, we’re
getting closer and closer to 20 with our 𝑓 of 𝑥 values. However, we never actually reach 20
from either side. From this, we can say that as 𝑥
tends to five, 𝑓 of 𝑥 approaches 20. If we convert this into math
language, then we reach our solution. Which is that the limit as 𝑥 tends
to five of 𝑥 squared plus three 𝑥 over the square root of 𝑥 minus one is equal to
20.

Let’s now consider how we can find
the limit of a function using a graph. Consider the following
examples.

If the graph represents the
function 𝑓 of 𝑥 is equal to 𝑥 minus three, determine the limit as 𝑥 tends to
negative one of 𝑓 of 𝑥. In order to find the limit of 𝑓 of
𝑥 as 𝑥 approaches negative one, we simply need to look at the graph. However, we first need to know what
we’re looking for. We can say that the limit as 𝑥
approaches negative one of 𝑓 of 𝑥 is the value that 𝑓 of 𝑥 approaches as 𝑥
tends to negative one. We need to consider the graph of 𝑓
of 𝑥 around 𝑥 is equal to negative one. We can see that just to the right
of the function, as 𝑥 approaches negative one, 𝑓 of 𝑥 approaches negative
four.

And now, we can consider 𝑓 of 𝑥
just to the left of negative one. We can see that the function is
doing the same, but in the opposite direction. It is tending towards negative
four. Therefore, using a wordy definition
of a limit, we can say that the limit as 𝑥 approaches negative one of 𝑓 of 𝑥 is
equal to negative four.

Now, we’ve seen a relatively
straight forward example of how we can use a graph to find a limit. Let’s consider a slightly less
obvious example.

Determine the limit as 𝑥 tends to
two of the function represented by the graph.

Now, if we look at the graph, we
can see that the function is called 𝑓 of 𝑥. And we’ve being asked to find the
limit of 𝑓 of 𝑥 as 𝑥 tends to two. In other words, this is the limit
as 𝑥 tends to two of 𝑓 of 𝑥. This can also be described as the
value 𝑓 of 𝑥 approaches as 𝑥 tends to two. So we need to find this value of 𝑓
of 𝑥 using our graph. We can consider what 𝑓 of 𝑥 is
doing around the value of 𝑥 is equal to two. We’ll need to consider 𝑓 of 𝑥 on
both the left and right of two. Let’s consider 𝑓 of 𝑥 on the
right of 𝑥 is equal to two. We can see that as 𝑥 gets closer
and closer to two, 𝑓 of 𝑥 is decreasing. And it is decreasing towards this
point here, which has a value of three. So we can say that as 𝑥 tends to
two from the right, the value of 𝑓 of 𝑥 approaches three.

Let’s now consider what happens on
the left of 𝑥 is equal to two. We can again see that as 𝑥 gets
closer and closer to two, the value of 𝑓 of 𝑥 is decreasing. And from the graph, we can see that
it is decreasing towards the same value that 𝑓 of 𝑥 is approaching from the right
as 𝑥 approaches two. And that’s a value of three. So now we can say that as 𝑥
approaches two from the left, 𝑓 of 𝑥 approaches three. Since 𝑓 of 𝑥 approaches the same
value from the left and the right of two, we can therefore conclude that the value
that 𝑓 of 𝑥 approaches as 𝑥 tends to two is three. And so, we reach our solution,
which is that the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to three.

In this last example, we’ve seen
how we can use a graph in order to find the value of a limit at a point, even if the
graph has a sharp turn at that point. Let’s now move on to another
example.

Determine the limit as 𝑥
approaches two of 𝑓 of 𝑥 if it exists.

Here, we have been given the graph
of 𝑓 of 𝑥, and we’re trying to find the limit of 𝑓 of 𝑥 as 𝑥 tends to two. If we try to find the value of 𝑓
of 𝑥 when 𝑥 is equal to two, we can see that 𝑓 is, in fact, undefined. However, this does not mean we
cannot find the limit. We know that the limit as 𝑥
approaches two of 𝑓 of 𝑥 is the value 𝑓 of 𝑥 approaches as 𝑥 tends to two. In order to find this limit, we
simply need to consider 𝑓 of 𝑥 around two not specifically at two. Let’s consider the 𝑥-values just
to the right and just to the left of 𝑥 is equal to two.

Let’s start by looking at 𝑓 of 𝑥
just to the left of 𝑥 is equal to two. We can see that as 𝑥 gets closer
and closer to two from below, the value of 𝑓 of 𝑥 is getting closer and closer to
three. And if we consider the 𝑥-values
just to the right of 𝑥 is equal to two, then we can see that as 𝑥 gets closer and
closer to two from the right, the value of 𝑓 of 𝑥 is decreasing and getting closer
and closer to three. Since 𝑓 of 𝑥 is tending to the
same value as 𝑥 approaches two from both the left and right and this value is
three, we can conclude that the limit as 𝑥 approaches two of 𝑓 of 𝑥 is equal to
three. In this last example, we’ve seen
how we still may be able to find the limit of 𝑓 of 𝑥 as 𝑥 approaches a particular
𝑥-value, even if 𝑓 is undefined at that particular 𝑥-value.

We will consider one final
example.

Using the graph shown, determine
the limit as 𝑥 tends to three of 𝑓 of 𝑥.

Here, we have the graph of 𝑓 of
𝑥. And we’ve been asked to find the
limit as 𝑥 tends to three. We can see that at 𝑥 is equal to
three, 𝑓 of 𝑥 is defined to be negative five. However, when we are finding the
limit of a function at a particular point, the value of that function at that point
does not matter. What matters is what’s happening to
the function around that point. This is because the limit as 𝑥
approaches three of 𝑓 of 𝑥 is defined to be the value 𝑓 of 𝑥 approaches as 𝑥
tends to three. Let’s consider what’s happening to
𝑓 of 𝑥 to the left and to the right of 𝑥 is equal to three.

If we consider 𝑓 of 𝑥 to the left
of 𝑥 is equal to three, we can see that 𝑓 of 𝑥 is increasing and getting closer
and closer to the value of two, as 𝑥 is getting closer and closer to three. And as 𝑥 approaches three from the
right, 𝑓 of 𝑥 is again increasing. And it is also getting closer and
closer to the value of two. Now, this tells us what we need to
know about 𝑓 of 𝑥 as 𝑥 tends to three, since from both the left and right, the
value of 𝑓 of 𝑥 approaches two as 𝑥 approaches three. And so, even though the value of 𝑓
of three is equal to negative five, the limit as 𝑥 approaches three of 𝑓 of 𝑥 is
equal to negative two, which is our solution to this question. In this previous example, we’ve
seen how even though a function may be defined at a different point at a particular
𝑥-value, the limit as 𝑥 approaches that particular 𝑥-value of 𝑓 of 𝑥 may be
different to the value of 𝑓 of 𝑥 at that point.

We have now covered a variety of
examples. Let’s look at some key points of
the video.

Key Points

The limit as 𝑥 approaches 𝑎 of 𝑓
of 𝑥 is the value 𝑓 of 𝑥 approaches as 𝑥 tends to 𝑎. When finding a limit using a table,
we consider the 𝑓 of 𝑥 values as 𝑥 gets closer and closer to 𝑎 from both above
and below. The value 𝑓 of 𝑥 approaches is
equal to the limit as 𝑥 approaches 𝑎 of 𝑓 of 𝑥. When finding the limit as 𝑥
approaches 𝑎 of 𝑓 of 𝑥 from a graph, we consider the values of 𝑓 of 𝑥 near 𝑎
to find the value 𝑓 of 𝑥 approaches as 𝑥 tends to 𝑎. This value is equal to the limit as
𝑥 tends to 𝑎 of 𝑓 of 𝑥.

One quick final point here is,
using tables and graphs, especially graphs, is a very visual way to find limits. And it can really help you to
understand what a limit of a function is. If a question has asked you to find
the limit of a function and not given you a graph of the function, you may find it
useful to draw a graph. And then, you should easily be able
to spot the limit of the function at that point.