[SOUND] Fundamentally, limits are promises.
When I tell you the limit of F of X equals L, as X approaches A; I'm
promising you something. I'm promising you that I can get F of X
as close to L as you like as long as X is close enough to A.
Thinking of limits as promises helps us to understand statements like these.
Let's suppose that you tell me that you know the limit of f of x as x approaches
a is equal to something, maybe this is equal to l.
And maybe the limit of g of x as x approaches a is equal to m.
What that really is, is a promise that you can make F of X as close to L as I
like. And it's a promise that you can make G of
X as close to M as I like, as long as I'm willing to make X close enough to A.
But if you can promise me that you can make F of X close to L, and G of X close
to M. Then I can turn back and promise you that
F of X plus G of X is as close to L plus M as you like.
Alright? If I want to make this close to
something, I just ask you to make F of X close enough to L, and G of X close
enough to M. So that F of X plus G of X, is as you
like to L plus M. In other words, the limit of a sum is the
sum of the limits. And the limit of a difference is the
difference of the limits. And the limit of a product is the product
of the limits. And what about quotient.
And something similar is true for division.
If the limit of F of X as X approaches A is L.
And the limit of G of X as X approaches A is M, which isn't zero, then the limit of
F of X over G of X, as X approaches A is L over M.
In other words, the limit of the quotient is the quotient of the limits, provided
those limits exist and, the limit of the denominator is non-zero.
Let us do something with our new found knowledge about limits of quotients.
Here's a limit problem: I'm going to limit x-squared over x plus one as x
approaches two, alright? I'll promise you that x-squared over x
plus one is close to something whenever x is close enough to two.
This is the limit of a quotient, and the limit of the quotient's the quotient of
the limits, provided the limit of the denominator is not zero, and in this
case, it's not. So the limit of the quotient is the
quotient of the limits. [SOUND] Here's the limit of the
numerator. The limit is X approaches two of X plus
one. This is the limit of the denominator.
Now this is the limit of x squared. X squared is X times X.
This is a limit of a product. And the limit of a products the product
to the limits so I can replace the limit of the numerator with a limit of X as X
approaches two times the limit of X as X approaches two.
because this is the limit of X times X and here's the product limits.
Limit of X times the limit of X as X approaches two.
The denominator here is the limit of X plus one as X approaches two but that's a
limit of a sum and the limit of the sum is the sum of the limits.
So the limit of X plus one [SOUND] is the limit of X plus the limit of one, as X
approaches two. Lets keep going.
So I've got the limit of X times the limit of X over the limit of X plus the
limit of one. And all of these limits are being taken,
as X approaches two. What's the limit of X?
That's asking what can you guarantee X is close to if you're willing to have X be
close enough to two? Two is the limit of X as X goes to two.
So limit of X as X goes to two is two. The limit of X as X goes to two is doubt
is for multiplying, divided by limit of X as X approaches two plus with the limit
of one as X approaches two. This is asking, what can I guarantee one
is close to two if I am willing to have X be close enough to two.
Well, one is already close to one, right. The limit of a constant is that constant.
So this is just one. Two times two is four.
Two plus one is three. And so the limit of this expression is
four-thirds. At this point you are asking yourself why
is the rule for limits of quotients different than limits of the products.
A limit of a product is a product of the limits.
Laws of limits exist. Why do I have to worry about the limit of
a nominator being non-zero, when I'm taking the limit of a quotient?
Most basically the problem is that you can't divide by zero.
You can't go around telling people that the limit of a quotient is a quotient of
the limits because the limit of the denominator might be zero and then you'd
be telling people to divide by zero, which they can't do.
You can't divide by zero. But you can think about it even a little
more subtlety. You know, let's kind of unpack this a
bit. Here's an example to think about: the
limit of x over x minus three as x approaches six.
This is no problem, alright? The numerator is a number close to six,
it's how we're thinking about it, and the denominator is a number close to six
minus three. A number close to six minus three, that
means the denominator is a number close to three.
Now, we've got a number close to six divided by a number close to three.
Well, that's a number close to two. And, indeed, I mean, this limit is equal
to two. I can make this quotient as close to two
as I like. Because I can make the numerator as close
to six as I need, the denominator is as close to three as I need, to guarantee
that this ratio is as close to two as you like.
So that limit is two. But what if instead of asking about the
limit as x approaches six, I'd ask about the limit as x approaches three.
Well, then what would I know? Then I'd know that the numerator was a
number close to three, and the denominator was a number close to three
minus three. The denominators aren't close to zero.
The limited denominator is zero. That's exactly the scenario that the rule
for taking limits of quotients is forbidding us from considering.
We're not allowed to use the rule for limits of quotients here because the
limited denominator is zero. But what really goes wrong?
I mean, yeah, I can't divide by zero. Fine, I'm not going to divide by zero,
I'm just dividing by numbers close to zero.
But what happens when I divide by numbers close to zero?
A number close to three divided by a number close to zero.
Is that close to anything? If a number's close to zero it might be
positive, and very small. Three divided by a small positive number
is a huge positive number. What if the denominator were a number
close to zero but negative? Very small, negative number close to
zero. Three divided by a small but negative
number? That would be a hugely negative number.
Every negative. Well, this means this thing should be
getting close to both a gigantic positive number and a very, very negative number.
This thing isn't getting close to anything.
Fundamentally that's the problem but not all is lost.
Think about this slightly harder example: the limit of x-squared minus one over x
minus one as x approaches one. It's a limit of a quotient.
So your first temptation is to replace the limit of a quotient, by the quotient
of the limits. But you can't replace it without quotient
of limits, because the limit of the denominator is zero.
The limit of X-1 as X approaches one is equal to zero.
So it seems like our limit laws have failed us.
And then we have got one more trick up our sleeve.
Look at the numerator. X squared - one that factors as X+1
times, X-1. First record that fact.
And I am going to record the fact that the numerator factor is X+1 times X-1.
Now I've still got a limit of a quotient and the limit of the denominator is still
zero. So it seems like we're stuck.
But now I've got a factor of X minus one in the numerator and a factor of X minus
one in the denominator, and I can use that fact.
What I want to do is imagine canceling these, right.
I'd like to write that this is equal to the limit of just X plus one, as X goes
to one. But note, these are not actually the same
function. Alright?
This thing up here is not defined at one, this thing is defined at one.
And yet the limit doesn't care. The limit only depends upon values of the
function near one. And near one, this and this are exactly
the same. I'm not allowed to plug one into this, I
am allowed to plug one into this. They're different functions.
But those functions are equal if you're only considering values that are near and
not equal to one. As a result, these limits are the same.
This is another limit of a sum, and the limit of a sum is the sum of the limits,
so this is the limit of X plus the limit of one as X approaches one.
The limit of X as X approaches one is one.
The limit of one, which is a constant, as X approaches one is one.
This is one plus one. This is two.
And so that limit is equal to two and our limit laws have again saved the day.
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