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The following table gives the values of the function π at several values of π₯.
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What does the table suggest about the value of the limit as π₯ approaches negative five of π of π₯?
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In this question, weβre given a table of values for a function π.
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In the top row of our table, weβre given our input values of π₯.
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And in the bottom row of our table, weβre given our outputs π of π₯.
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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₯.
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To do this, letβs start by recalling exactly what we mean by the limit as π₯ approaches negative five of π of π₯.
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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.
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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.
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And we can do this from our table.
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Letβs start with values of π₯ greater than negative five.
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Letβs start with the first column in our table.
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Our input value of π₯ is negative 4.895, and we want to see what our output value is.
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We can see that π evaluated at this value of π₯ is negative 14.01, and this is useful to know.
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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.
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So, weβre going to want to pick values of π₯ even closer to negative five.
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Letβs look at the third column in our table.
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Now, we can see the input value of π₯ is negative 4.979.
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And this is even closer to negative five than the value of π₯ in our first column.
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And one way of seeing this is to calculate the difference between these two values.
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We can see when we subtract negative 4.979 from negative five, we get a value closer to zero.
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So, letβs see if our values of π of π₯ are getting closer to our value.
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We can see now that π evaluated at this value of π₯ is negative 14.003.
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And now, weβre starting to see a pattern.
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In our first value of π₯, our output was negative 14.01.
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However, now, weβre even closer to negative 14 with a value of negative 14.003.
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And once again, we can explicitly calculate how close weβre getting to this value of negative 14.
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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.
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And this is, of course, much closer to zero.
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In our table, we have one more value of π₯ which is even closer to negative five from the right.
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So, we can see if this pattern continues.
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This time, weβre going to use the value of π₯ is negative 4.9999.
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For due diligence, weβll start by checking that this value of π₯ is indeed closer to negative five.
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We can see the difference between this value of π₯ and negative five is much closer to zero than the others.
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So, it is indeed closer to negative five.
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Now, all we need to do is check that our output is closer to negative 14.
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We can see this directly from our table.
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However, we could also calculate this difference directly, and we see that itβs equal to a value of 0.0001.
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And now this confirms our pattern.
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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.
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So, it appears we want to choose our value of πΏ equal to negative 14.
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However, we need to be careful.
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Remember, we always need to check what happens from both sides.
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So, we also need to see what happens from the other side.
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We need to choose our values of π₯ less than negative five.
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We can do this in exactly the same way.
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Letβs start with the last column in our table.
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Thatβs when π₯ is equal to negative 5.02.
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This time, we can see that our output will be negative 13.895.
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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.
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And we could check this directly.
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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.
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So, we need to see whatβs happening to our values of π of π₯ as we move along the columns.
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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.
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And one way of doing this is to check the difference between our outputs and negative 14.
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In the final column of our table, our output value of π of π₯ is negative 13.895.
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So, the difference between this and negative 14 is calculated by negative 14 minus negative 13.895.
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And we can calculate this is equal to negative 0.105.
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We can do the same for the second to last column in our table.
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We get an output value of negative 13.92.
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And we can calculate the difference between this and negative 14 is negative 0.08.
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And we can do exactly the same for our last two columns.
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We get the differences of negative 0.002 and negative 0.001.
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And once again, we can see a pattern in these values.
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Theyβre getting closer and closer to zero.
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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.
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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.
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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.
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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.