Video: Finding the Limit of a Function from Its Graph at a Point of Removable Discontinuity If the Limit Exists

Find lim_(π‘₯ ⟢ βˆ’3) 𝑓(π‘₯).

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Video Transcript

Find the limit as π‘₯ approaches negative three of 𝑓 of π‘₯.

We’re given a sketch of the function 𝑓 of π‘₯. We need to determine the limit as π‘₯ approaches negative three of 𝑓 of π‘₯. First, we need to remember what this means. As our values of π‘₯ are getting closer and closer to negative three, we want to find out what our outputs of the function 𝑓 of π‘₯ we’re approaching. So let’s start by looking at negative three on our π‘₯-axis.

The π‘₯-axis represents the inputs of our function. And we know the 𝑦-axis will tell us the outputs for those values of π‘₯. And at this point, we might be worried about a problem. We can see that our function 𝑓 of π‘₯ is not defined when π‘₯ is equal to negative three. This is represented by the hollow circle.

But we need to remember when we’re taking the limit as π‘₯ approaches a value of a function, we don’t need to worry about the value of our function at this point. We only care what happens around this value. So let’s see what happens when our values of π‘₯ approach negative three from the left. This means our values of π‘₯ will be less than negative three.

As our values of π‘₯ approach negative three from the left, we can see that our outputs are getting closer and closer to nine. And this is because this point has 𝑦-coordinate nine. So our outputs are getting closer and closer to nine. But what happens when our values of π‘₯ approach negative three from the right?

That means our values of π‘₯ will be bigger than negative three. We can see a very similar situation happens. As our values of π‘₯ approach negative three from the right, we’re getting closer and closer to 𝑦-coordinate nine. This means that our outputs are also approaching nine.

Therefore, as we approach negative three from either the left or the right, our outputs are getting closer to nine. This means we can say the limit as π‘₯ approaches negative three of the function 𝑓 of π‘₯ is equal to nine.

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