Question Video: Finding the Limit of a Function from Its Graph Mathematics • Higher Education

Determine the limit as ๐‘ฅ โŸถ โˆ’3 of the function represented by the graph.

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Video Transcript

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

Weโ€™re given a sketch of a function. We need to use this to determine the limit as ๐‘ฅ approaches negative three of this function. Letโ€™s start by calling our function ๐‘“ of ๐‘ฅ. This means that our curve is given by ๐‘ฆ is equal to ๐‘“ of ๐‘ฅ. Letโ€™s now recall what we mean by the limit as ๐‘ฅ approaches negative three of a function ๐‘“ of ๐‘ฅ. The notation for this is the limit as ๐‘ฅ approaches negative three of ๐‘“ of ๐‘ฅ. And what we mean by this is the value that ๐‘“ of ๐‘ฅ approaches as ๐‘ฅ tends to negative three.

In other words, as our input values of ๐‘ฅ are getting closer and closer to negative three, we want to know what happens to our outputs ๐‘“ of ๐‘ฅ. 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 ๐‘ฅ-axis, letโ€™s mark ๐‘ฅ is equal to negative three. Remember, we want to know what happens to our outputs as ๐‘ฅ is getting closer and closer to negative three. So letโ€™s start by looking at what happens as ๐‘ฅ gets closer and closer to negative three from the left.

If we input ๐‘ฅ 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 ๐‘ฅ 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 ๐‘ฅ 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 ๐‘ฅ is equal to negative three, our output is equal to negative two.

But remember in our definition, when we say ๐‘ฅ tends to negative three or ๐‘ฅ is approaching negative three, we mean that ๐‘ฅ is getting closer and closer to negative three. ๐‘ฅ is never equal to negative three. In other words, it doesnโ€™t matter what happens when ๐‘ฅ 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 ๐‘ฅ 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 ๐‘ฅ is equal to zero, we can see that ๐‘“ of zero is equal to one. And if we input ๐‘ฅ 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 ๐‘ฅ 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 ๐‘ฅ approached negative three from the left, our outputs approached negative two. And as our values of ๐‘ฅ approached negative three from the right, our functionโ€™s 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 ๐‘ฅ approaches negative three of the function given to us in the graph is equal to negative two.

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