### Video Transcript

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

For part i of this question, weβre
being asked to find the value of π of zero. Which is to say, we must evaluate
our function when π₯ is equal to zero. We can graphically find the value
of π of zero in the following way. We take the point on the π₯-axis
where π₯ 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 π₯ is equal to zero, that would mean our function
would be defined here instead. However, we do not see a solid dot
anywhere when π₯ is equal to zero. Which must mean that our function
is undefined when π₯ is zero. Given this fact, we cannot assign a
value to π 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 π₯ approaches zero of π of π₯?

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 π of π₯ will approach πΏ as the value of π₯ approaches π from
both sides. But weβre not concerned with the
point where π₯ is equal to π. Okay, We want to find the limit as
π₯ approaches zero of our function. In other words, the π in the
general form of our limit equation is zero. Okay, to find the value of our
limit, we need to find πΏ. Perhaps weβll call this πΏ one to
be clear. πΏ one is the value that π of π₯
approaches as π₯ approaches zero from both sides. To find this πΏ one, we can look at
our graph and see what happens to our curve as the value of π₯ approaches zero.

We see that as π₯ approaches zero,
π of π₯ approaches one. In other words, weβre 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 π₯ which are arbitrarily close to zero but not where π₯ is actually equal
to zero. Great, we have found that π of π₯
approaches one as π₯ approaches zero from both sides. Which means, the value of πΏ one is
equal to one. We can now rewrite our limit
equation in full. The limit as π₯ approaches zero of
π of π₯ is equal to one. We have now answered both parts of
our question.

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