In this video, I will introduce the limit laws:
a few theorems about limits that we use very often.
I want to explain what they say and how to use them,
but ALSO why we care about them.
Before I tell you what the limit laws are,
I want to begin with a motivating example:
I want to prove
that in order to calculate the limit of ANY polynomial at ANY real number,
I simply have to plug it in; to evaluate at that number.
For example, if I want to calculate lim_(x → 2) (x^4 - 3x),
I simply plug x = 2, evaluate, and THAT's the answer.
I have proven this for a couple of examples in past videos,
and any individual proof of a specific polynomial can be done,
although sometimes it gets messy.
But I don't want to have to do this one polynomial at a time!
I would like to write a proof for ALL polynomials at once,
and then from there on,
every time I need to compute the limit of a polynomial,
I plug it in, knowing that has been rigorously proven.
Can I do this easily?
Yes, I can. Here is the plan:
I'm going to proceed in two steps.
First, I'm going to prove some basic limits.
The ONLY limits I have to prove from the definition
are these two very easy functions:
the function f(x) = x, and a constant function.
That's it.
Why these two?
Because they are the building blocks of polynomials;
every polynomial can be written
as sums and products of these two functions.
So after proving this basic limit,
I will prove that sum and product behave well with respect to limits,
and that's what the limit laws say.
Specifically, the limit laws tell me that IF I have two functions f and g
and I assume they have a limit at the same point a;
the limit of f is L and the limit of g is M,
THEN the sum must also have a limit; it will be the sum L + M,
and similarly, the product must also have a limit;
it will be the product L ⋅ M.
And there is a third limit law about quotient
that I won't use in this particular case, but is also true.
The limit law for the quotient requires that we do NOT divide by 0.
And that's it! That's all I need.
I need to prove the value of two limits from the definition,
and I need to prove this theorem about sums,
and this theorem about products.
How will this help me?
How can I go from just THIS
to concluding that the value of the limit of ANY polynomial
is just the same as plugging it in, rigorously?
Well, let's look at an example.
Let's say I want to calculate lim_(x → 2) (x^4 - 3x).
This function is the sum of two smaller functions: x^4, and -3x.
So using the limit law for sums,
I can write this limit as the SUM of TWO limits.
And now, EVERY piece is the product of functions.
-3x is -3 ⋅ x, and x^4 is x ⋅ x ⋅ x ⋅ x.
So using the limit law for product,
I can write each of these limits as a product of limits.
And look! I am now down to the basic limits:
the limit of x, and the limit of a constant.
So I can use that result to plug it in, and end up with the value of the limit.
That's it! And the exact same process is going to work for ANY polynomial.
So my plan works!
If I manage to prove the value of the basic limits
and I manage to prove the limit laws,
I will NEVER have to use the (ε, δ) definition
to calculate the limit of a polynomial again!
I will know that I just have to plug it in,
and that will HAVE been rigorously proven.
This, by itself,
is enough justification for why we care about the limit laws.
It is worth it to spend a little bit of time
trying to PROVE this theorem,
because it's going to save us SO MUCH more time in the future.
But to be fair, this is just ONE example of an application;
the limit laws are SO useful,
and we use them all the time in so many circumstances.
But before I get carried away,
the limit laws come with a VERY IMPORTANT warning:
the limit laws ONLY APPLY if I assume the initial limit exists.
In the statement here, I am ASSUMING f and g have a limit,
and then I'm DRAWING conclusions about the sum and product;
if I don't KNOW that the initial functions have a limit,
I cannot draw conclusions.
For example, lim_(x → 0) x = 0, lim_(x → 0) 3/x does not exist,
but their PRODUCT HAS a limit.
So 0 ⋅ (DNE) = ... who knows? Could be anything.
Or in this other example,
the function 1/x does NOT have a limit at 0;
the function (2x - 1)/x does NOT have a limit at 0,
but their SUM HAS a limit.
If this confuses you, pause the video
and write a little bit of the algebra, and then it will become clear.
So, once again,
in order to draw conclusions about the sum or product functions,
I need to KNOW in advance
that each function individually has a limit.
If I don't know that,
I cannot draw ANY conclusion, neither positive nor negative.
Okay, so what about the proofs?
I'm not going to prove the basic limits; I'm going to leave those as an exercise.
As far as (ε, δ)-proofs go, these two are the simplest there are.
The examples of limits of polynomials from the definitions
that I've computed in past videos were harder.
As for the limit laws,
I'm going to write the proof of one of them as an example.
So in the next video,
I will write a proof of the limit law for sums.