Aadil V.

asked • 02/15/22

Use this table of value to define the limit lim 𝑥→2 𝑓(𝑥) if possible. Explain your thinking.

x = 1.99 x =1.999 x = 1.9999 x=2 x = 2.0001 x=2.001 x=2.01
y = 1.9701 y=1.997001 y = 1.99970001 ??? y = 2.00030001 y=2.003001

y=2.0301 (for rightmost bottom cell, not able to type in it)

Mark M.

Did you do any thinking?
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02/15/22

Heidi A.

Aadil, as you read the y-values from left to right until the empty space and then from right to left towards the empty space, does it seem like the numbers are approaching a specific number that should go in the empty space?
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02/16/22

Aadil V.

Ohh, I see what you mean, so it should be y=2?
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02/16/22

Paul M.

tutor
This question is misleading. The question suggests that the limit is 2. But without more knowledge about f(x) a conclusion about the limit cannot be drawn. What if f(x) is discontinuous?
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02/16/22

Aadil V.

@Paul M. Should I put the limit does not exist then? To me it seems like the limit is 2
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02/16/22

Stanton D.

Paul M. -- The limit may still be 2, with the function lacking that specific point. The limit is a value, NOT a point. Aadil -- yes.
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02/16/22

Paul M.

tutor
Stanton D. I agree that the limit may still be 2...but it may not be...and with the information given the limit cannot be determined. Yes, the limit is a value specifically defined when (in this case) x is arbitrarily close to 2...and in this problem there is not sufficient information to conclude that limit=2.
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02/16/22

Andrew F.

Paul M. , Thank you so much for emphasizing the correct mathematics here. This type of problem is common in some textbooks and comes out of the Harvard "reform" movement--the idea is for students to use a table of values to "see" the "as x approaches "a," then f(x) approaches "L," but this approach is not mathematically defensible even if well-intentioned. No information about a limit can be found by using a table. A table can help illustrate a known limit, but because there is no "next number," we have no idea what happens if all we have is a table.
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02/17/22

1 Expert Answer

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Christopher W. answered • 02/20/22

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F(x) essential represents the y value of the function when a particular x is substituted into the function. So the table's x values are substituted and the y value beneath represents the answer. A limit is simply asking what does the function value approach, so we can see that no matter side of 2 that we choose to approach form the answers is close to 2.

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Andrew F.

Christopher W., I think you should consult the comments to this question--tutor Paul M. has the mathematically correct response even if the textbook or software or teacher is looking at the problem along the lines you suggest. --Andrew F.
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02/20/22

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