### 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.