Video: Finding the Limit of a Function from Its Graph If the Limit Exists

Using the graph shown, determine lim_(π‘₯ β†’ 3) 𝑓(π‘₯).

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Video Transcript

Using the graph shown, determine the limit as π‘₯ tends to three of 𝑓 of π‘₯.

Here, we have the graph of 𝑓 of π‘₯. And we’ve been asked to find the limit as π‘₯ tends to three. We can see that at π‘₯ is equal to three, 𝑓 of π‘₯ 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 π‘₯ approaches three of 𝑓 of π‘₯ is defined to be the value 𝑓 of π‘₯ approaches as π‘₯ tends to three. Let’s consider what’s happening to 𝑓 of π‘₯ to the left and to the right of π‘₯ is equal to three.

If we consider 𝑓 of π‘₯ to the left of π‘₯ is equal to three, we can see that 𝑓 of π‘₯ is increasing and getting closer and closer to the value of two, as π‘₯ is getting closer and closer to three. And as π‘₯ approaches three from the right, 𝑓 of π‘₯ 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 π‘₯ as π‘₯ tends to three, since from both the left and right, the value of 𝑓 of π‘₯ approaches two as π‘₯ approaches three. And so, even though the value of 𝑓 of three is equal to negative five, the limit as π‘₯ approaches three of 𝑓 of π‘₯ 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 π‘₯-value, the limit as π‘₯ approaches that particular π‘₯-value of 𝑓 of π‘₯ may be different to the value of 𝑓 of π‘₯ at that point.

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