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Copy pathU0S3V05 Limit Law for Division, Part 3.txt
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U0S3V05 Limit Law for Division, Part 3.txt
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#
# File: content-mit-18-01-1x-captions/U0S3V05 Limit Law for Division, Part 3.txt
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# Captions for MITx 18.01.1x module [3VysBc12dOI]
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# This file has 48 caption lines.
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# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
All right.
Let's finish this third part of the limit law for division.
We're dealing with the case where
the limit of the denominator is 0
and the limit of the numerator is 0, but remember
that we're not going to be saying that this limit equals
0/0.
When x is near a, the numerator and denominator
are approaching 0, but that doesn't mean that they equal 0.
It's just that they're close to 0.
So the quotient here is going to be
some small number over some small number when x is near a.
And the interesting thing about this case
is that just knowing that the numerator and denominator are
small doesn't tell us anything about how big this quotient is.
What we really would need to know
is how small the numerator and denominator
are relative to one another.
For instance, the numerator could be 0.01.
That's pretty small, but the denominator
could be even smaller.
It could be 0.00001.
And when you divide this, you get 1,000.
Or it could be the reverse.
It could be that the numerator is 0.00001 and the denominator
is 0.01.
And now the quotient becomes 0.001 or 1/1,000.
In the example from the last video that we did,
we were taking the limit as x approaches 0 of 2x over x.
Here the denominator is always exactly half the size
of the numerator and so we always got a quotient of 2.
The bottom line is that if the denominator and the numerator
are both going to 0, the limit of the quotient
could be anything.
It could be big.
It could be small.
It could be somewhere between any of that.
Now this doesn't mean that we just give up and say,
I don't know, but it does mean that we
have to do more work in order to determine
what the limit is in each case.
We have some questions for you.
And then in the next video, we'll
discuss just what sort of work you can do in order
to figure out limits like this.
OK.