So now we know the definition of Multivariate function
and know how to draw some understanding of it by administering the levels of its function to the graph, right? So now let's talk about the limits of the
multivariate function. Let us just stick with our basic understanding
of what the limit is. The limit is the value that our function resembles the most. Basically, what it means
for our multivariate case, in our multivariate
case we can just generalize our
extremely complicated but a useful definition with Epsilon Delta
technique, right? So what have we
been saying in it? We are basically saying
that the way you see is called Functions Limit. In some for example,
two-dimensional points whatsoever. If for any given deviation
we can go as close to limit point that
function do not deviate from our limits sufficiently, but much more than our
undefined deviation, right? Okay. That's basically
what it is and it doesn't change at all but as a phrase, we can go as close
to our limit point implies that we now can say what is close to how
limit point and what is not. So basically what
we're talking about; we're talking about
our point a b, we are talking about some
arbitrary point x y, and we're going to say is
that the distance between these two points are
smaller than our, for example, our delta z. It is delta neighborhood where in the close neighborhood
of our limit point. In order to do it, we need
to understand some basics. For example, how to calculate the distance between
point x and the y, and a and b. In order to do it,
we're just going to use straight-out pythagoras
theorem which is well, you don't know what to do,
use pythagoras theorem. So our distance here is
basically square root from sum of squares of the
sides of this triangle. Which means that the
square root from x minus a squared plus y minus b squared. Okay. That's nice but how does it differ from the case of
single-headed functions? From the standpoint
of definition, it's pretty much the same. But let us consider for
example our real plan and we are looking at some
of our limits point ab. So what should we expect? Previously what we've
talked about is that we're just trying to get as close to our limit point from all possible straight
directions, right? You do remember we
had two definitions, one complicated
with epsilon delta and one with sequences, right? So sequences basically told us go to the limit points as
close as you can and then the limit of the
sequence of actions failures should approach all the
same value c, right? Because this can actually be easily extrapolated here but, and this is a very big but here. Now they're considering all
approaches on the plane, not on our green straight
line or our real numbers. Okay? So now we can
go for example, from the left, from the right
as in single variate case. But also vertically from the top or by some straight
line here or maybe just by parabola or just by some random points into the plan. So the only rule here we need to approach our limit point ab. Okay? So this is extremely more complicated because we have additional degree of freedom,
additional variable y. Thus we need to consider much more possible
approaches that we can use to get to the limit
function, limit point. More importantly, the idea is that we need
just to consider limit by one variable hands-on but the other will not suffice by the very idea that there's only one way to get here, right? Lets write english way. Firstly, you need to throw yourself on to the for example, x axis or y axis and then you need to
consider only one variable. This is only two approaches, we need to consider them all. So let us revise what
we do have here. Since we've established
that we need to consider all approaches to prove for example or
calculate our limits, we need to prove that
on all approaches, we get the same results. It's nightmarish. So the technique we're going to use primarily in
order to calculate function limits is basically to truncate this calculation to the case of single variables. For example, change all the
calculations we're doing into the case of single
variate functions by substituting variables, by grouping variables,
by just subtracting and just turning our function into the pieces of
single-variate functions at all. Okay, So but to prove the
absence of the limit, it's quite nice to
consider that we can just demonstrate a
pair of approaches. For example, horizontal or
vertical one or some others with two different limits and so the limit itself won't exist. So for example, let us take a look at basic
polynomial functions. Do not be disturbed
by the last term. As you can see, it's
basically a polynomial just we can get rid out of brackets and thus we will get pretty nice
polynomial function. How can one say that the limit of this function
is easy to calculate? Because it should be easy to
calculate, let's face it. What if the polynomial function is not easy to calculate, right? First of all, all
arithmetic rules are the same because the
definition is the same, right? Thus, we should consider
only the limits of the terms separable right? So in order to consider the limits of this
term for example, let us take the term five y
powered five multiplied by x. Okay? Why it is easily truncatable to the case
of single varied function? Because it's easier as a product of single
varied at function y equals to function
equals to five, function equals 2y powered five. Its single-varied function
towards y, right? As a product of function x which is single varied
function towards x. Thus it approaches,
five approaches five, y approaches one thus y powered five approaches one power five and
10x approaches too. This term approaches ten. Okay? That's easy and
thus we can calculate the limit of this polynomial
just variable vice. To proceed further,
we will cancel with some more complicated examples
in the following video.