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Hi there.

In the last two sessions of two very important

We define the concept and simple We can consider applications made.

These partial derivative and integral multi-storey concepts.

As you will remember these four basic process is used.

But an important operation as well as their There, it also limits the process.

In this section, this limit a little more importance I would like to highlight.

Of functions of one variable and two

transition to variable function I want to tell.

We can ask the following question:each function of each

differentiation and integration can be calculated where Did you?

It also functions of one variable in We know everywhere derivative calculated.

But now many of the applications in this derivatives calculated.

But this derivative to calculate what conditions

that it is possible, when you can not do We know better.

Something similar in the integral.

Therefore, the general answer to this question is no.

However, in the majority of cases required continuities,

finite values of the function defined As a result to be calculated.

So how can answer this question?

With the concept of limits.

So as you can see anywhere in the limit is required.

Four operations since the beginning of telling 're doing.

Differences, to calculate changes minus

subtraction and division operation, or are used together.

Here, we obtain the derivatives exchange As that process.

As a collection of processes that accumulation

and impact achieved by using a combination of It is.

But these four operations is not much of a limit There is also not the place.

Do you think, from first grade From these four are dealing with the transaction,

Did you spend a lot of time means that the limit let's spend a little time for.

This limit usually related to these issues Students do not like.

But I try not to love you because I'm going to dry

instead defines them functionally I thought, how can import.

But you also want to do a trial

for a good faith understanding Waiting effort.

If you do not learn it now forever with a small account

There was also the philosophy of making the limit You will be but practical calculations

because it will help you in making computer made in the accounts,

In numerical integral derivative a common practice to calculate the fine.

Here infinitesimal thoroughly zero can come

remains finite, but with a small account finite 're doing.

Therefore, this limit to understand

in computer software programs to use

well, in order to develop yourself will be a useful thing.

Now I'm not here

Instead of simple conceptual definitions, such as to describe them

I'll try a single variable functions

Basic question:selected a small epsilon and when

can be made as small as desired, a Delta is

When, I wonder where x minus x is zero, so the We choose smaller than epsilon.

I.e., x is zero near a x function values

The difference value a l could provide

until there is a LAN that can provide problem, we are looking to answer the question.

So I'm this close to x to zero

I've decided that one of the thousands I'm going to close.

I wonder if changing delta of the epsilon depending on

a function near x, x

values near zero values

l Can I closeness?

If such a line of work there are limits and we say this,

l f x x x goes to zero in the limit we say.

After a bit of it visually I'll explain.

Immediately following this limit to exist and

a concept that we use a lot of continuity concept.

Continuity of water.

If there is this limit, ie, x x goes to zero f x is the limit of

If, that is, if there is such a line and this line x is zero to the value of the function

x is equal to zero in this kind of functions is continuous call.

That is the essence of continuity, actually.

There are two limits open one here one written, but written

we say that x x goes to zero limit,

We know that x is zero for him We write here explicitly.

Equal to the limit of the function as wonder Is it?

So you see here the limit order changes here before

establishing the value of the function with respect to x I would take the limit.

Here, the x on the right side of the x

taking the limit to zero when We calculate the value of the function.

This means that the continuity of the limit order can be changed.

If you think this limit as geometric if so,

If there is this line, this line will be calculated how?

Now these questions.

It says that if the function l

If the value of x is zero, this function is said to be continuous.

That there is continuity and limits The easiest way to understand look at reverse.

Problems can arise when?

Some are very simple.

Y is equal to the square root function, e.g. x we take.

This x-axis with the horizontal axis of symmetry the following parabolic conflicting.

Let me say it to you in a negative Find the value.

Does a minus in the definition?

Because outside the domain of the function involved.

Function definition means initially in the region should be.

Function is outside the domain of negative the square root of a real one

will be worth, we also

If we work with real functions is undefined for us.

A second undefined type, this more often I've encountered something.

This is completely in accordance with intuition.

I wonder if there is such a leap in function Does this give us a limit?

Limits were saying the following:x around epsilon

we choose, so here is a narrow strip we choose.

I wonder if it's a line in this strip Can you find value?

Around this time, a delta de la I choose by selecting a horizontal strip.

I can stay in this lane.

See, you see, I go to the right, to say that the limit is zero.

Because of symmetry seems reasonable therefore, zero.

I go right, I'm a little bit around it

I'll choose a narrow strip of the delta thick.

As you can see I went a little bit to the right pat, it went off the road.

I go a little left out of this strip again I went.

Come on, let's try other values not zero.

Where you try, the function this value, you can select

s to confine in narrow strips near not possible.

There is no limit to His call it x In this jumping-off point is equal to zero.

Similarly seçsek such a function, a divided by x squared

zero as x squared minus x, ie, x is zero Near a function tending to infinity.

Now here is there a limit?

Now let's say you chose such a line x

remain near zero for x is equal to the epsilon I choose to.

I wonder if it's as much as I want a delta

Playing can be trapped in this narrow strip Can I?

See, here I say a little bit to the right limit when you get to pat

I went off of it, it points near the small strip.

Rather, it went to the left of x is zero again I went out of limits that you see

You can not find a line like that again function dispenser into this trap.

Take as another type of a function x minus

x x squared minus x is zero divided by zero square.

See if x is equal to x is zero zero divide this otherwise undefined zero.

These two things to get rid of undefined You can.

You can exclude this function, here, here You can bring a discontinuity.

Now making similar calculations in this wherein

You can see the algebraic limit as x x0 squared minus the square

x minus x minus say x0, x0, x0 x plus the ie 0/0

When you purify it makes the term You can find a limit.

But as it is, 0/0 is uncertain.

In order to make certain that's it We use limits.

Again, he had a kind.

Sine 1 / x you say x approaches 0 As you can see, this

sine of the variable 1 / x, supremely fast is changing.

This is a result removed from the computer.

This function approaches zero such infinite time starts to oscillate.

You do not even know where.

Here are possible.

You are changing a little bit right now going on a very big difference.

On top of this there is no limit at x = 0 we say.

Single undefined function, the function is undefined,

there is no limit, i.e. the limit is mean both

values will be determined individually, and As such there will be a certain finite.

Other specific examples You can find them in, but the main types are.

Function may be undefined, and the limit Things could not be found.

A lot of variables go up and down There is a similar definition.

So, before a slight modification of two Let's start from variable.

He said in one variable x is near 0 x of the

let's choose a small strip consisting of an infinite, Let's select range.

Let's call it epsilon.

You're doing the same thing here, only here

There are a vector in space x, which defines 0.

This vector points near receive.

This limits you except a

giving circle.

Everywhere in this apartment You can.

We do not want you to go to the border because it smaller than epsilon

We also get to an open area is going on.

Again we say in one variable functions f (x) - l, wonder

By playing with such a strip of delta Can we stay inside?

In multivariate same thing.

I wonder, is there such a line with delta playing function

changing x and y in which Can we find a line that keeps?

So as you can see the definition of highly parallel.

Thus shows only difference.

When you get a point x0 in one dimension.

Near this point right or left

You can approach but on a line will remain.

There are two types ie orbit,

Thank you from left to right will close will close.

However, in two dimensions in the plane of the point x0 point.

Around its x - x0 vector length

Get a circle as it turns out, at the border You do not want to stay.

It's a circle with radius epsilon in the

from any point x to point x0 will close.

As you can see on the right two kinds way, if there is a right

to approach a left and right to approach the You have to stay on.

Here are forever kind of way.

Here we give some examples.

For example, you can come along the curve, head on

You can come to the right along a straight line.

You can come along such a curve or delta ago

x can take it to zero, then the delta y You can take it to zero.

So for these broken lines You can approach to x, x0 to.

Or you can take before the delta y to zero.

Then you can take the delta x to zero.

Infinitely many orbits is more like it clear.

This is a fundamental difference there.

Again, these results all along this trajectory

I wonder when you reach the value of x0 is s

next function value of a value if you can imprison,

with delta to remain in the vicinity playing.

Here are two kinds orbit You can have,

with infinitely many orbits here you can have.

Continuity, but still the same, if these limits If this limit function

If the estimated value of x0 and y0 here function of two variables constantly call.

The only variable was the same, where yi think of x, x0 when he went to the limit

If this limit function in x0 If the value is obtained, you can see

concepts such as a single application of the size difference is there.

Were doing in practice univariate

Is there no limit to leap in function Do you look for

approaching from the left or right of x0, x0 Did you find approached the same value?

If you find these values continuity and assume that there were limits.

So left and right limits equal at that point so that the limit

We decide that, and the function constantly that we decide.

In multivariate functions x0, y0 to

any nearby trajectory can have.

They are looking at the most basic right in is the orbit.

See the first force, where x and y a first force

The slope m and y point, yx

xy x0y0 to point or point We arrive.

Indeed y = y0 when x = x0 is going on.

If it shows concretely, for example, As this is a point xy

x0y0 point from which the slope m arrive with the right.

Of course, the problem that this change m these will be

're getting the same value to the value of different We arrive?

Here it serves to distinguish them calculation.

Let's take an example here.

Very often encountered, in applications encountered mathematical

in terms of a single species, but a very common 0/0 is an issue.

Sometimes it happens sometimes it does not limit.

Is there a limit do not see here?

That once the uncertainty 0/0 occurs.

And this for several orbits My vara.

An orbital inclination m in length, which I vara.

here

17:46

Here you can start with positive values, but here

trilogy four orbits roughly the same thing.

Or a zero in orbit before x We had made zero then y are doing.

As it orbits.

See detailed one here Let's examine.

This orbit like that.

Concurred in orbit before x zero

are doing by y plus You are approaching.

If we went back to them before x 0 y / y stayed square, thus 1 / y.

y is always a plus side we approach this year

plus valuable, but a shrinking points.

So this ratio will always be valuable plus but increasingly

means that the limit will be growing at a value will have forever.

Although we have approached for the second orbit We are here again x 0.

x and hence goes.

Again y / y at square 1 / y remains.

But we're approaching from the negative side wherein the absolute value

but with the growing minus sign Number encounter.

This absolute limit remained at minus sign value goes to infinity.

As you can see for these two trajectories of two We have kind of problem.

Once you are unable to finite values, From a further aspect

In reaching minus infinity to plus direction We arrive forever.

Here these trajectories other such we can examine the way.

For example, we take y = mx, so that third orbit Something like that.

Here we reset x y de reset.

So if we put y mx instead, see here x

plus mx, x as the denominator common factor If we take

1 m times x plus there, plus a 1 m in the denominator There is a square.

Does that square again by one plus x squared There are x's more simple.

There are two kinds of problems here.

M For a time-dependent finite x value

We find, also goes to zero at x the same

Direction x plus zero as above

While three of which went for x length plus values approaching zero.

Here we find positive infinity.

If we do the same operation on four accounts

Also, this time we're getting negative infinity.

There are two kinds of problems as you can see.

Both bonded to M., x remains nonzero as long as

Both in one plus one minus infinity Coming forever.

There are other examples as you homework I have some left.

For example, let us consider the following two functions.

Sinus x plus y divided by x plus y or and

the second as a function of x plus y x minus y.

Now x is equal to 0 in the limit of y is 0 there Do you?

0 of 0 divided by x we put directly take So

that there is an ambiguity, here divided by 0 0.

Now this year to eliminate is equal to m *, write,

As before, such a trajectory that Let us draw near to the center for.

Does one plus x As you can see above, one plus m x below.

If you say one plus m xa, as you can see Split sinus is dying.

If you approach from the approach of the x plus

by or Less to 0 by 're approaching sinus

divided by approaching the limit u u0 and we find that there is one.

So here's a limit function is there.

Otherwise, if we do the other in the second example As you can see we do

m below the above plus 1 minus 1 m You can not define an end to occur.

And according to the different values of m different, You can find values for this function.

For example, if m is 0 m equals 0 If sinus

x divided by x, x 0 to 1 when that limit We know that.

But if m 1 divided by 2 and, where 1 plus 1

divided by 2 to 3 divided by 2 x happens, in the denominator

in 2 divided by 1 minus 1 would be 1 divided by 2x.

This limits the time you get to the x 0 when

for example, x 3 with Multiply Share on here,

3, in the denominator, multiply the share hit a 3x 3 you will find.

If u 2x divided by 3 is the same here as well but a third remains as one can find.

As you can see here, where Depending on how close

The value of the function are different, one or 3.

No eternity here, but from each other different this ik,

The limit of this function even enough To say that.

But see if you put m is 1 m is equal to the denominator of the first sheep going 0

More here without having to take x to 0 an infinity occurs.

then if x 0 in the ass again remains forever.

There are two kinds of problems here.

It differs from others as well as endless.

But it any grain any iii shows that there is double the limit.

So even if finite, different values Because there are no limits.

I want to give you these papers.

I wonder if x is 0 and y is 0, this Is there a limit function?

See here differ from those in other above

wherein a first degree function of the

the square root is taken as a first-

In the event an effective There are functions.

Here, too, a little different from öbürki x y is 0 is 1.

Wherein x is 0 and the gene

y is equal to 0 if there are limits on See the call.

In the latter case is a limit Please select a call.

You can see these limits would lead to 0.

Now I wanted to make this theoretical part Limits you too

I hope that the coming dry with Description conceptually and in calculations by

Recalling also encountered situations And always something to be removed therefrom derivatives

not be integral, but still our In our example

So you'll encounter during your studies very

'll encounter in practice and in life applications

Generally, with the continuous function

to limit certain functions that We're trying.

So far, very few applications have made.

So we saw the partial derivative calculation.

Simple integral, integral two-storey I have seen calculations, but this is not enough.

The next two in our derivative

calculation methods and applications We'll see.

Integral calculations in subsequent sections methods and practices to be seen.

Therefore, to make this account to derivatives Our ability to account

develop and where they is applied, for example, three values, i.e.

in the calculation of minimum and maximum We will see applications.

Partial nature of the basic equations are expressed in derivatives.

For example, most of us know fluids There mechanics

There are gas dynamics, solid bodies, There strength.

Propagation of waves in electromagnetism There, in the electrostatic

There are heat conduction problems.

These are the basic equations of nature.

And in the event of an atomic scale Is that defines the Schrödinger equation.

But it in the first seven weeks We'll see.

In this course, following this second lesson

Two Functions of Several Variables in that We'll see.

But it is an important issue.

Also integral calculation methods and We need to consolidate applications.

The main idea we found so far, but We improve our skills account yet.

They are very mixed accounts on the computer will do.

But, to be held in the computer accounts to make

I got to know these methods are for.

Analytically on the computer to know many will not be able to make creative process.

Goodbye for now.

Hopefully then this calculation

You will see the methods, concepts also We will further consolidate.