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.