U0S1V13 Limit Laws.txt

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So far, we've discussed limits of a single function
at a particular point.
In this video, we'll talk about what
you can do if you know the limit of multiple functions
at the same point.
So here we have the limit of f of x as x approaches a.
And let's say that's equal to 5.
And then, over here, we have the limit
of g of x at the same point a.
And let's say that we know that that limit equals 3.
And then we can talk about combinations
of the two functions.
For instance, we could take f of x plus g of x.
And we can try to find the limit of this sum at the same point,
so as x approaches a.
And it turns out that as x approaches a,
the limit of the sum is equal to the sum of the two limits--
in other words, 5 plus 3, or 8.
Let's go ahead and justify this.
We're looking at what happens as x approaches a.
And we know that f of x is approaching 5.
Now, that doesn't mean that f of x ever actually equals 5.
But it'll be off by just some small error.
So we can write f of x equals 5 plus this little E. And E
stands just for small error.
This is actually the Greek letter epsilon.
And I'm going to call it epsilon sub 1,
since we're going to have another one for g.
g of x, we know, is going to be 3
plus some small error of its own,
which I am going to call epsilon sub 2.
And we know that as x tends to a, these small errors,
epsilon 1, epsilon 2, they're going
to be getting smaller and smaller, going to 0.
Meanwhile, f of x plus g of x, just add this and this,
and it'll be 8 plus epsilon-1 plus epsilon-2.
So it'll be off from 8 by epsilon-1 plus epsilon-2.
But of course, both epsilon-1 and epsilon-2,
those are our small errors.
They're getting really, really, really tiny
as x gets close to a.
So their sum is also going to get really small.
And that means that this error goes to 0.
And the limit of f plus g is going
to be 8, just as I promised.

And this works in general, as well.
So I'll erase this first.
And then the general statement is,
if the limit as x approaches a of f of x equals some number L,
and the limit as x approaches a of g of x
equals some number M, then the limit
as x approaches a of the sum, f of x plus g of x, is L plus M.
So the limit of the sum is the sum of the limits.
And this will work with one-sided limits,
as well, as long as you're coming in from the same side
on all three limits.
For instance, x approaches a from the right,
a from the right, a from the right, like that.

This is called the limit law for addition.
And a similar thing will hold for subtraction of functions.
If we have these limits of f and g,
then the limit as x approaches a of the difference,
f of x minus g of x, will be the difference
of the two limits, L minus M.
Same for multiplication.
The limit as x approaches a of the product, f
of x times g of x, will be the product of the two limits, L
times M. So those are the limit laws for subtraction
and multiplication.
There is a limit law for division.
But it's a bit more complicated.
If you take the limit as x approaches
a of the quotient, f of x over g of x,
it will L over M if the bottom limit M is not 0.
So far, so good.
But if M does equal 0, then things
become really interesting.
I'm not going to say exactly what happens just yet.
But it's so interesting and so important to calculus
that we're going to save that discussion
for a whole other section.
So stay tuned.