### Video Transcript

Determine the limit as ๐ฅ approaches negative three of the function represented by the graph.

Weโre given a sketch of a function. We need to use this to determine the limit as ๐ฅ approaches negative three of this function. Letโs start by calling our function ๐ of ๐ฅ. This means that our curve is given by ๐ฆ is equal to ๐ of ๐ฅ. Letโs now recall what we mean by the limit as ๐ฅ approaches negative three of a function ๐ of ๐ฅ. The notation for this is the limit as ๐ฅ approaches negative three of ๐ of ๐ฅ. And what we mean by this is the value that ๐ of ๐ฅ approaches as ๐ฅ tends to negative three.

In other words, as our input values of ๐ฅ are getting closer and closer to negative three, we want to know what happens to our outputs ๐ of ๐ฅ. Since we want to know what happens as our inputs are getting closer and closer to negative three and we know that our inputs are represented on the ๐ฅ-axis, letโs mark ๐ฅ is equal to negative three. Remember, we want to know what happens to our outputs as ๐ฅ is getting closer and closer to negative three. So letโs start by looking at what happens as ๐ฅ gets closer and closer to negative three from the left.

If we input ๐ฅ is equal to negative seven, we can see that our function outputs negative six. If we were to get even closer to negative three and input ๐ฅ is equal to negative five, we can see that our function outputs negative four. We can get even closer by inputting negative four. We see that our function will output negative three. And we want to know what happens as our inputs of ๐ฅ get closer and closer to negative three. And we can see that as our inputs are getting closer and closer to negative three from the left, our outputs appear to be getting closer and closer to negative two. In fact, in this case, when ๐ฅ is equal to negative three, our output is equal to negative two.

But remember in our definition, when we say ๐ฅ tends to negative three or ๐ฅ is approaching negative three, we mean that ๐ฅ is getting closer and closer to negative three. ๐ฅ is never equal to negative three. In other words, it doesnโt matter what happens when ๐ฅ is equal to negative three. It only matters what happens around this value. But this is only one side of the story. What happens as ๐ฅ approaches negative three from the right? This time, our inputs will be greater than negative three. We can do this using a similar method.

For example, if we input ๐ฅ is equal to zero, we can see that ๐ of zero is equal to one. And if we input ๐ฅ is equal to negative two, we can see that our function outputs negative one. So ๐ of negative two is equal to negative one, and we can do exactly the same argument. What happens as our values of ๐ฅ get closer and closer to negative three? And once again, we can see exactly the same thing happens. Our outputs are getting closer and closer to negative two.

So as our values of ๐ฅ approached negative three from the left, our outputs approached negative two. And as our values of ๐ฅ approached negative three from the right, our functionโs outputs also approached negative two. So we can just say that this limit is equal to negative two. Therefore, we were able to show the limit as ๐ฅ approaches negative three of the function given to us in the graph is equal to negative two.