So let us consider more constructive examples of
limits of single-variant functions and
the cases of its absence.
So first, let's start with the ideas that we sometimes
can just separate variables into singular functions.
Sometimes we need to group them.
For example, considers the limits
when x and y approaches point zero
a and we need to find limit of
sine of products x and the y divided by x.
So, of course, one thing that we can always do,
we can decide whether there
is or not the in-determinant form.
But it's quite easy because the product of
x and y approaches zero multiplied by a approaches zero.
The sine of zero is zero,
thus we divide zero by zero and this
is definitely in-determinant form.
So we need to do something with it.
Firstly, the thing that it resembles
is our second important limits,
which was sine t divided by t. It
approaches one as the argument approaches zero.
So we need to somehow come up with the same concept here.
Basically, let us assume that since
we do not know any formulas from
trigonometry which separates the sine of products,
thus we need to group
the variables in the sine into new variable
t. Thus if we
find this t variable integrated in the denominator,
then we can get our second important limit here.
In order to do it, we can just simply add y multiplier
and divide by a multiplier our main denominator.
What we're looking at, we are looking at
sine of t divided by t as it approaches one,
multiplied by function of one variable y as it
approaches a by our task.
Thus, our limit equals to a. That was easy.
Let us consider this nice view on
the surface and we're actually talking about
limit at this very point.
This is zero and this is our z axis,
so this is a nice point.
As always, I've attached this graphs
into additional materials where you can
rotate it and play with it after the lecture.
So this was a nice one,
a rather simple example.
Let us move to more frightening one.
Okay. This is the function of n we're
considering the limit at point 0,0 and
it is the division of
two polynomial functions which should be nice
obviously because in singularity function
it's always one of the simplest cases.
So what happens if we
consider xy divided by x squared plus y squared case?
Well, basically speaking this is nightmare and I'm going
to show you in the following manner.
First, let us assume that we need to have
some understanding of what it can be in the results.
So we need to find any possible candidate for a limit.
In order to do so, we consider
some direction of approach of 0,0 point.
For example, I'm going to
consider the horizontal approach
where x equals some parameter t and y equals to zero.
Thus, we need to find the limit when t approaches zero.
So let us substitute x and y with our directions.
Thus, we get t multiplied by
zero divided by t squared plus zero squared,
which basically means that we are
considering limit of zero.
Not infinitesimal function but zero itself,
and the limit of zero is simply zero.
Okay, that's nice.
The same applies for a vertical direction, for example.
But if we just play alone with direction,
for example, if equals to t and y equals to t,
this diagonal direction, thus we
get pretty much different function as a result
because it results into t multiplied by t
divided by t squared plus t squared as t approaches zero.
Or in other words, one-half.
Since one-half is not zero,
this function doesn't have a limit at this very point.
Let us see what it basically means for our surface.
Basically, for our surface, let us look.
It means that here we have a vertical line.
In other words, if we just consider
direction x equals to t and y equals to for example kt,
which is any straight direction towards 0,0 point,
then our function is going into kt squared
divided by k squared t squared plus t squared,
or k divided by k squared plus one which is a constant.
Thus, all these directions are our levels.
Or basically what we're looking at,
we are looking at function with such nice raise as
levels with zero level on
the axis and one-half on the center.
Basically, our levels grows here,
but at the 0,0 point there is all possible levels.
Basically we are looking at function which resemble
all possible constants from zero to one-half,
which implies that there is
no one constant to rule them all, there is no limit.
Okay. So polynomial divided
by polynomial can be nightmarish.
But that means that it is always like this.
Well, it doesn't.
Consider that we increase the power
of polynomial in the denominator and then
maybe here we can get our nice answer.
Okay. Firstly, the thing here is to
call the limit because if we just consider
our horizontal approach or just a common case because we
have a polynomial of higher order in nominators and
denominators as by our consideration
of little on notation,
we basically get that this nominator
is a little or with regard to denominators,
it's limit should be zero.
Well, in the horizontal notation it basically proves
our point because our functions turns into a zero itself.
How one should prove that the function approaches zero
since it's hard to move towards a single variate keys,
they are going to write the following.
We are going to find the boundary
on the absolute value of
this function which is single variant function.
For example, assume that we expelled
from the denominator plus y squared,
thus denominator necessarily becomes
a low value because y squared is non-negative.
Thus, if denominator is lower then the whole fraction is,
it becomes greater which is x squared multiplied
by y divided by x squared or just y approaches zero.
If y approaches zero,
basically our function is bounded by a y and minus y.
All of it approaches zero,
thus our function approaches zero.
Okay. We proved it and that's nice.
Let's take for example,
look at the surface.
It's kind of bulk,
pretty much awesome surface.
It's not extremely damaged,
not extremely complicated, just nice curve of paper here.
Okay. As the last trick that I'm going to show you here,
let us consider pretty much the same function
but in order to
prove that something is zero,
sometimes people use so-called polar coordinates.
Which basically implies that we can rewrite
our x and y coordinates in terms
of the distance to zero
and an angle of this for example x-axis.
Thus, x turns into r cosine
of the angle and
y turns into r multiplied by sine of the angle.
Our idea is basically that we have
our function minus it's possible limit should
be bounded by some function which depends on
the distance towards the limit function
and it approaches zero.
It's basically the definition of our limit,
it means that the deviation from the limit of
our function tends to be less and
less if it's closer and closer to our limit point.
So let us check it for our function.
Basically what we are looking at,
we are looking at r power three
multiplied by cosine squared multiplied by sine squared,
which doesn't actually match because we
are going to take up slot value and
sine cosine function are bounded by absolute value 1.
Denominator is basically r
squared minus possible limit is zero.
Thus, we get after dividing nominator denominator
by r squares we get r multiplied by cosine and sine.
In order to expel the angle,
expel the direction just to leave
understanding that we depend only
on how close we are to the limit point,
we're going to say that
both cosine and sine functions are not greater than one.
Thus, it's all bounded by the distance
by r and r approaches zero.
As r approaches zero,
thus basically we've proven the definition of the limit
and thus we've proven
that the limit of our function is zero.
It is sometimes nice trick when
you see for example this expressions,
x squared plus y squared reaches r squared,
a lot in your function or for example in the denominator.