# Question Video: Finding the Area of a Region Bounded by a Given Curve Mathematics • Higher Education

The following figure is the graph of the function π, where π(π₯) = sin π₯/π₯. i) What is the value of π(0)? ii) What does the graph suggest about the value of lim_(π₯ β 0) π(π₯)?

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### 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, weβre 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 πΏ as the value of π₯ approaches π from both sides. But weβre 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 πΏ. Perhaps weβll call this πΏ one to be clear. πΏ one is the value that π of π₯ approaches as π₯ approaches zero from both sides. To find this πΏ 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, weβ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 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 πΏ 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, letβs 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, weβve seen in this question that our function does not even need to be defined at a point where a limit is being taken.