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Question Video: Finding the Limit of a Function from Its Graph at a Point of Removable Discontinuity Mathematics • 12th Grade

Determine the limit of the function as 𝑥 ⟶ −2.

02:28

Video Transcript

Determine the limit of the function as 𝑥 approaches negative two.

We’re given a graph which contains the curve which represents a function. We need to determine the limit of this function as 𝑥 approaches negative two. Since our 𝑦-axis is labeled 𝑓 of 𝑥, this function will be called 𝑓 of 𝑥. Let’s start by recalling what we mean by the limit of a function as 𝑥 approaches negative two. We write this as the limit as 𝑥 approaches negative two of 𝑓 of 𝑥. And this is the value that our outputs of 𝑓 of 𝑥 approach as 𝑥 approaches negative two.

In other words, as our input values of 𝑥 are getting closer and closer to negative two, we want to see what happens to our outputs. We know our outputs are on the 𝑦-axis and our inputs are on the 𝑥-axis. So, let’s mark 𝑥 is equal to negative two on our 𝑥-axis. To see what happens as 𝑥 is approaching negative two, let’s start by looking at what happens as 𝑥 approaches negative two from the left. This means our inputs will all be less than negative two. Let’s start with 𝑥 is equal to negative four.

When we input 𝑥 is equal to negative four, our output value is five. 𝑓 of negative four is equal to five. Let’s now pick a different point. What if we’ve chosen 𝑥 is equal to negative three? We can see that our output value is two. 𝑓 of negative three is equal to two. But we want to know what happens as our values of 𝑥 get closer and closer to negative two. And now, we can see something interesting. We can see from the hollow circle on our curve, our function is not defined when 𝑥 is equal to negative two.

But remember, when we say that 𝑥 tends to negative two or 𝑥 approaches negative two, this means our values of 𝑥 is never equal to negative two. In other words, when we’re finding limits, we don’t care what happens at this value; we only care what happens around this value. And we can see from our sketch, as our values of 𝑥 are getting closer and closer to negative two from the left, our outputs are getting closer and closer to one.

But we can also ask the question, what would have happened if our values of 𝑥 had approached negative two from the right? In fact, we would get a very similar story. 𝑓 of zero is equal to five; 𝑓 of negative one is equal to two. And as our values of 𝑥 get closer and closer to negative two from the right, we can see our outputs are getting closer and closer to one. So, in both cases, as our values of 𝑥 approached negative two, the outputs of our function approached one.

This means we’ve shown the limit as 𝑥 approaches negative two of the function 𝑓 of 𝑥 is equal to one.

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