### Video Transcript

The following table gives the values of the function π at several values of π₯. What does the table suggest about the value of the limit as π₯ approaches negative five of π of π₯?

In this question, weβre given a table of values for a function π. In the top row of our table, weβre given our input values of π₯. And in the bottom row of our table, weβre given our outputs π of π₯. We need to see if we can use this table to determine information about the limit as π₯ approaches negative five of π of π₯. To do this, letβs start by recalling exactly what we mean by the limit as π₯ approaches negative five of π of π₯.

We recall we say that the limit as π₯ approaches negative five of a function π of π₯ is equal to some constant value of πΏ if the values of our outputs of π of π₯ are approaching πΏ as the values of π₯ are approaching the value of negative five from both sides. So, to check if the limit given to us in the question is indeed approaching some finite value of πΏ, we need to see if our outputs of π of π₯ are approaching some value of πΏ as the inputs of π₯ are approaching negative five from both sides. And we can do this from our table. Letβs start with values of π₯ greater than negative five.

Letβs start with the first column in our table. Our input value of π₯ is negative 4.895, and we want to see what our output value is. We can see that π evaluated at this value of π₯ is negative 14.01, and this is useful to know. But remember, to see what happens to our limit, we want to know what happens as π₯ is getting closer and closer to negative five. So, weβre going to want to pick values of π₯ even closer to negative five.

Letβs look at the third column in our table. Now, we can see the input value of π₯ is negative 4.979. And this is even closer to negative five than the value of π₯ in our first column. And one way of seeing this is to calculate the difference between these two values. We can see when we subtract negative 4.979 from negative five, we get a value closer to zero. So, letβs see if our values of π of π₯ are getting closer to our value.

We can see now that π evaluated at this value of π₯ is negative 14.003. And now, weβre starting to see a pattern. In our first value of π₯, our output was negative 14.01. However, now, weβre even closer to negative 14 with a value of negative 14.003. And once again, we can explicitly calculate how close weβre getting to this value of negative 14. Calculating the difference between these outputs of π of π₯ and our value of negative 14, we can see we go from a value of 0.01 to a value of 0.003. And this is, of course, much closer to zero.

In our table, we have one more value of π₯ which is even closer to negative five from the right. So, we can see if this pattern continues. This time, weβre going to use the value of π₯ is negative 4.9999. For due diligence, weβll start by checking that this value of π₯ is indeed closer to negative five. We can see the difference between this value of π₯ and negative five is much closer to zero than the others. So, it is indeed closer to negative five.

Now, all we need to do is check that our output is closer to negative 14. We can see this directly from our table. However, we could also calculate this difference directly, and we see that itβs equal to a value of 0.0001. And now this confirms our pattern. As our values of π₯ are getting closer and closer to negative five from the left, our outputs of π of π₯ are getting closer and closer to negative 14. So, it appears we want to choose our value of πΏ equal to negative 14. However, we need to be careful. Remember, we always need to check what happens from both sides. So, we also need to see what happens from the other side. We need to choose our values of π₯ less than negative five. We can do this in exactly the same way.

Letβs start with the last column in our table. Thatβs when π₯ is equal to negative 5.02. This time, we can see that our output will be negative 13.895. Remember, to check the value of this limit, we need to see what happens as our values of π₯ are getting closer and closer to negative five. And we could check this directly. However, we can see directly from our table that our values of π₯ are indeed getting closer and closer to negative five as we move along the columns.

So, we need to see whatβs happening to our values of π of π₯ as we move along the columns. We want to check that these outputs are getting closer and closer to negative 14 because then weβll be able to conclude that this limit is equal to negative 14. And one way of doing this is to check the difference between our outputs and negative 14. In the final column of our table, our output value of π of π₯ is negative 13.895. So, the difference between this and negative 14 is calculated by negative 14 minus negative 13.895. And we can calculate this is equal to negative 0.105.

We can do the same for the second to last column in our table. We get an output value of negative 13.92. And we can calculate the difference between this and negative 14 is negative 0.08. And we can do exactly the same for our last two columns. We get the differences of negative 0.002 and negative 0.001. And once again, we can see a pattern in these values. Theyβre getting closer and closer to zero.

So, this means as our values of π₯ got closer and closer to negative five from the left, our outputs are getting closer and closer to negative 14. Therefore, as our values of π₯ are approaching negative five from both sides, our outputs of π of π₯ are getting closer to negative 14. And this is exactly what we say in our definition to say the limit of π of π₯ as π₯ approaches negative five is equal to negative 14.

Therefore, by looking at the values of our function π in this table, we were able to show that the table suggests that the limit as π₯ approaches negative five of π of π₯ should be equal to negative 14.