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

Determine the limit as π‘₯ ⟢ 2 of the function represented by the graph.

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Video Transcript

Determine the limit as π‘₯ tends to two of the function represented by the graph.

Now, if we look at the graph, we can see that the function is called 𝑓 of π‘₯. And we’ve being asked to find the limit of 𝑓 of π‘₯ as π‘₯ tends to two. In other words, this is the limit as π‘₯ tends to two of 𝑓 of π‘₯. This can also be described as the value 𝑓 of π‘₯ approaches as π‘₯ tends to two. So we need to find this value of 𝑓 of π‘₯ using our graph. We can consider what 𝑓 of π‘₯ is doing around the value of π‘₯ is equal to two. We’ll need to consider 𝑓 of π‘₯ on both the left and right of two. Let’s consider 𝑓 of π‘₯ on the right of π‘₯ is equal to two. We can see that as π‘₯ gets closer and closer to two, 𝑓 of π‘₯ is decreasing. And it is decreasing towards this point here, which has a value of three. So we can say that as π‘₯ tends to two from the right, the value of 𝑓 of π‘₯ approaches three.

Let’s now consider what happens on the left of π‘₯ is equal to two. We can again see that as π‘₯ gets closer and closer to two, the value of 𝑓 of π‘₯ is decreasing. And from the graph, we can see that it is decreasing towards the same value that 𝑓 of π‘₯ is approaching from the right as π‘₯ approaches two. And that’s a value of three. So now we can say that as π‘₯ approaches two from the left, 𝑓 of π‘₯ approaches three. Since 𝑓 of π‘₯ approaches the same value from the left and the right of two, we can therefore conclude that the value that 𝑓 of π‘₯ approaches as π‘₯ tends to two is three. And so, we reach our solution, which is that the limit as π‘₯ approaches two of 𝑓 of π‘₯ is equal to three.

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