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.