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

Video Transcript

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.