So recently, we've discussed the case where two functions
relate to each other as a constant multiplier.
But sometimes, it's nice to understand that one function,
for example, decreases sufficiently faster,
grows sufficiently faster than others.
For example, consider the case of x squared,
and x around zero.
If you do remember,
graphs, basically x squared,
and x do relate in the following manner,
x squared decreases to zero as x approaches zero,
much more faster than x itself.
In a way, it is sufficiently faster,
that means that it cannot be truncated to the case
of faster than some finite number of times, as in big-o.
So we can say that it's a different relationship if it is
relationship which is called little-o
or infinitesimal towards some other function.
It is here. There is a nice picture of our x,
x squared, and x power 3.
One function is called little-o
infinitesimal towards the other,
if its relation approaches 0
as x approaches our limit point
D. That's our formal definition.
In other words, one function is infinitesimal function,
Alpha multiplied by that. That's quite easier.
In case of x squared,
it's basically x multiplied by infinitesimal function x.
That's nice. Let us consider some examples.
For example, pretty much the same thing,
nothing ever changes here.
Start with two polynomials. What we should decide?
We should decide whether or not
their relation approaches 0 as for example,
let's start with X approaches 0.
Does this relation approach 0?
Let us write it in a manner X power n minus m,
and it approaches 0 in the case if n is greater than m,
because if this power is negative,
then we have the same sine
as 1 divided by X, that approaches infinity.
So this case is solved,
but what about X approaches infinity?
That's basically vice versa,
all the same sine,
this power should be negative in order that to
be 1 divided by X or so on,
which approaches 0 as X approaches infinity.
Going to stress out the folders.
There is no less or equal or greater and equal.
Here these are strict inequalities.
That's basically the difference between
little-o and big-o in terms of polynomial.
As for our example here,
we are going to rule of with an idea of
sine of X and X as it approaches 0.
So what we do know?
We do know that sine of X divided by X should
approach 0 in order
for it to be infinitesimal one toward the other.
Sine towards x.
But as you can clearly see by
our second most important limit, it approaches 1.
So thus if X approaches 0,
sine of X is not infinitesimal towards X,
it's equivalent towards the X.
Well, it's built in that.
But in case X approaches infinity,
that's the limit we have covered
recently, you do remember.
This is bounded function,
sine multiplied by infinitesimal function 1 divided by X,
thus the limit equals to 0,
and little-o notation holds.
Sine is infinitesimal towards X
if X is unbounded. That's nice.
Since we know the definition of little-o notation,
let us revisit it.
Firstly, what we stated?
We stated that f is
little-o towards g or f is infinitesimal towards g,
if it's a relation approaches 0, or in other words,
f is a result of product of
some infinitesimal function and g function.
So let us consider this following puzzle.
What does it mean if some function
is little-o towards the constant function one?
As for complex approaches A,
it doesn't actually matter for our case.
It basically means that this function f is
the result of product of the
infinitesimal n function 1, constant 1.
That basically means that our function is infinitesimal.
Written here. But also,
let us assume the following case.
Assume that our function f has a limit,
for example, capital A is the point A.
Thus, the function f minus A,
our capital A. F minus its limit
is infinitesimal by our arithmetic rules,
yes, X approaches limit point.
Thus, we can write it down that
f minus A is little-o towards 1,
or in other words,
f is its limit plus some infinitesimal function.
That's all that, but that's a nice notation.
Why? Because let us consider, for example,
our second important limit here.
Basically, using our just established relation
between function limit and little-o notation,
we can derive it in the manner that sine x divided by x
equals to 1 plus little-o towards 1.
That's nice.
But then you can just rewrite it in the following manner,
let us multiply both sides of the equation by X here.
Thus, we get sine X equals to X plus.
Here comes the trick.
Formerly, we should write X multiplied
by little-o towards 1.
But you do remember,
little-o towards 1 is
basically any infinitesimal function.
But what is the product
of infinitesimal function and some other function?
It is infinitesimal function towards the function itself.
Basically, here, it's written little-o towards X.
So let's just interchange it.
In other words, multiplication
in terms of little-o notation,
it's naive entity, you can just multiply it,
not the function itself,
but just the argument of little-o notation.
So why does it help us?
Let us rewrite this relation in terms of half an angle.
For example, sine of X divided
by 2 is X divided by 2 plus, once again,
we need to formally
write little-o towards X divided by 2,
but since we are talking about infinitesimal functions,
division by 2 doesn't change the fact that we
are looking at infinitesimal function.
It's actually going to meet any constant that appears
in infinitesimal notation here,
and just write down little-o towards x.
So remember, basic trigonometry,
1 minus cosine of X is 2 sine squared of half an angle.
Let us substitute our sine of half an angle
with our fancy little-o notation.
Thus, we get the cosine function is 1 minus 2,
X divided by 2 plus little-o towards X squared.
Or in other words,
1 minus X squared divided by 2 minus,
that's going to be extraordinarily complicated,
but we are going to live through it.
Just hold on. First of all,
we need to write carefully down what we're getting here.
We should get 2 multiplied by X divided by 2,
multiply it by little-o towards X.
So we get 2X little-o towards X,
minus 2 little-o towards X squared.
So let us spend some time with the last two terms here.
Basically, [inaudible] is constant multiplier,
which we do know doesn't matter,
we can just admit it to say equal for little-o notation.
We are looking at a multiplication by X,
we'll just establish that can easily
be put into the argument of little-o notation.
Thus, this term is
basically infinitesimal towards X squared.
Same applies in the case of
the second term in the following manner.
If I have two function infinitesimal towards X,
thus it's basically infinitesimal function
multiplied by X squared.
Thus we get infinitesimal
squared multiplied by X squared,
or simply infinitesimal towards X squared. It's easy.
So as a result,
we get 1 minus X squared divided by 2,
plus little-o towards X squared.
This is the asymptotic understanding of
cosine function in the neighborhood
of X, it's basically 1.
Then we get our square quadratic function. That's nice.
For some of you who know,
that's basically the start of
the channel series for the cosine function.
Done extremely,
easily extremely fast for the little-o notation.