AP Calculus AB The Limit of a Function

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to start talking about the limit of a function.0004

Very important topic, it is absolutely the foundation of all of calculus.0008

Let us jump right on in.0014

In the first few lessons, I have mentioned that given some f(x), the derivative involves some limit process.0016

I think I will work in blue today.0026

In the first few lessons, I mentioned that given f(x),0036

the derivative which we symbolize with a little symbol, the derivative f’(x) was found as follows.0054

The limit as h approaches 0, let us go ahead and put f’(x) is equal to quotient f(x) + h – f(x)/ h.0075

Basically, once we form this quotient, we simplify it.0094

Once we form this quotient and simplify algebraically, or whatever else that we need to do,0100

that is going to be algebraically, we get some function, let us call it g(x).0117

When we form this quotient, we simplify it and it gives us some function of x.0128

Then, we take the limit, then we apply this thing.0137

We end up taking the limit as h goes to 0 of some g(x).0146

More generally, we want to be able to handle things like this.0157

More generally, when we take a derivative, we are going to be taking limits.0162

But that is not the only time that we are going to be taking limits.0169

We want to be able to handle limits whenever they come up, not just in the context of differentiation.0172

Or generally, we want to be able to handle any function, the limit as x approaches some number of any function f(x),0178

whenever it might come up, whether it is in the context of derivative or not.0202

This is what we want to know about and this is what we are going to do for the next couple of lessons.0205

What this says is as follows.0212

Let me rewrite it so we have it on this page.0224

The limits as x approaches a of f(x).0227

This symbol says given some function f(x), what happens to this f(x),0231

as x itself gets closer and closer to some number a, a can be infinity, x could go off to infinity.0254

We ask ourselves, if x is going to infinity, what is the function doing?0274

As x approaches 5, what is the function doing, how was it behaving?0278

Is it oscillating, is it going off to infinity itself, is it getting close to a number?0283

That is what we are asking.0288

In other words, y = f(x).0289

We know the when we have f(x) = x², y = x².0302

f(x) and y are essentially synonymous.0306

For y = f(x), again, what this says, what the symbol says is, as x gets closer and closer to a,0310

does y get closer and closer to some number itself?0328

Does it go off to infinity?0346

Does y go off to positive or negative infinity?0352

Does y oscillate back and forth between two numbers?0358

Does y oscillate?0364

Does y bounce back and forth between two numbers?0375

What does y do, that is what we are asking, that is what this symbol says.0386

As x gets closer to some number, what does f(x) do?0391

Notice what happens to y?0398

As it turns out, all three of these things happen.0403

Sometimes y gets close to a number, sometimes it goes off to infinity, positive or negative infinity.0406

Sometimes it just oscillates between two numbers.0411

It turns out, all of these happen depending on the function and depending on the number you are approaching, depending on a.0417

Depending on what f(x) is, depending on x and f(x), and depending on a.0432

I'm going to describe this limit concept.0451

I'm going to describe this limit concept by looking at the graphs of several functions.0455

Because I want you to have an intuitive understanding of what the limit means.0488

I want you to have an intuitive understanding,0498

I think understanding is probably not the word I want to use.0517

I want you to have an intuitive feel for what the symbol limit as x approaches a of f(x) means.0519

If you have an intuitive understanding then any math that we do on a formal level will make sense.0534

If we just throw out some definition, some formal definition involving mathematical symbols,0541

we can explain but you need to have to get a feel for it.0549

It is very important.0552

The intuitive is actually more important than the formal mathematical of this level.0553

As a matter of fact, let me go ahead and take just a second to discuss this notion between formal and intuition.0560

Nowadays in calculus courses, in AP calculus, I believe in most classes, when they talk about this idea of a limit,0568

they introduce the formal definition of a limit.0576

You may or may not do it in your class, I’m not exactly sure.0581

It is my personal belief that at this level, when you are just doing calculus, multivariable calculus, linear algebra, and differential equations,0584

your first exposure to these things, you should not be exposed to formal definitions and what we call epsilons and deltas.0591

It is more important that you understand what is happening intuitively,0600

so that you can actually manipulate your mathematics, based on what you understand.0604

Rather than trying to fiddle with really intricate formal definitions.0609

For those of you that go on to mathematics, you are going to end up taking a course called analysis0614

where you go back and you revisit calculus.0619

But instead of actually doing computational problems,0621

you actually prove why it is that you can do the things in calculus that we are going to do, for this entire course.0624

You are going to see the formal definition.0632

It is going to depend on your teacher, the extent to which they actually want to emphasize it.0636

But it is my personal belief that it does not belong at this level.0640

For that reason, I'm not going to present a formal definition.0644

In some of the problems that I do later, I may mention it in passing.0647

But I want you to have the intuitive feel, before we do a formal definition.0653

Let us look at the following function.0658

Let us look at y = x² and ask the following question.0662

What is the limit as x approaches 3 of f(x) or same thing, because f(x) is x², what is the limit as x approaches 3 of x²?0676

What we are asking is, as x gets close to 3, what does x² do?0702

Write that down.0709

What we are asking is, as x gets closer and closer to 3, what is happening with y?0713

What is happening to y?0736

You know the answer already, but let us take a look at it.0740

Here is the function y = x² and here is our value of 3, right over there.0745

The limit, this symbol, the limit as x approaches 3 of x² is actually 2 symbols.0755

There are two things going on here and we have to deal with both.0764

That is fine, I will just do it here.0769

The first one is, what is the limit as x approaches 3, when you see a little negative sign to the top right of that number,0772

it means what is the limit as x approaches 3 from below 3?0778

In other words, 1, 2, 2.5, 2.6, 2.7, from the negative, from the bottom, of x².0783

We write that as the limit as x approaches , with the + from above 3, 5, 4, 3.5, 3.4, 3.1, as you get close from above.0793

This we call approaching from below.0810

This we call approaching from above.0820

When you see the symbol, limit as x approaches some number,0828

if it is not specified, whether you are approaching that number from below or from above,0831

you have to assume that you are approaching it from both.0836

You have to actually solve two limits, every time that is the case.0839

If they specify ready what the ±, then you just have to solve that one limit.0842

This is also called the left hand limit, this is called the right hand limit.0847

Left hand limit because you are approaching the number from the left.0850

In other words, you are approaching 3 from the left.0854

Right hand limit, you are approaching 3 from the right.0857

That is all that means.0862

Let us see what is happening.0865

Let us do this one over here.0868

The limit as x approaches 3 from the left, 2, 2.5.0871

It looks like as we get close to 3, y itself, what is y doing?0874

We just follow the path.0882

It looks like it is getting close to 9.0888

Now let us see what happens as x approaches 3 from above.0894

As x approaches 3 from above, this way, the function itself y, it looks like the y value is also approaching 9.0897

We did from below, it looks like it is approaching 9.0911

It looks like both from above and from below, this is 9 and this is 9.0915

As x approaches 3, it appears that f(x) or y is approaching 9.0927

That is what our little arrow means.0941

Arrow means it is approaching 9.0943

Let us confirm this with a table of values.0946

We have the graph, the graph is one way to actually deal with a limit.0948

Let us see what happens, let us see what the graph does.0952

It will tell us something about what is happening.0955

Let us confirm this with an actual table of values.0958

Here is the graph and here is the table of values.0963

Here, from here to here, here is x approaching a from below.0967

Notice 2.5, 2.7, 2.9, 2.99, 2.999, 2.9999.0975

When we say gets closer and closer, that is really what we mean.0986

We mean this part right here and it gets really close.0989

We see that as it approaches 3 not equals 3, as it approaches 3, the function y which is x² is going 6.25, 7.29, 8.41, 8.9, 8.99, 8.999.0994

Yes, it looks like it is approaching 9 from below.1010

From above, this is x approaching 3 from above, 3.5, 3.3, 3.1, 3.01, 3.001, 3.001.1014

You notice the y values, they descend and they come down to about 9.1032

Sure enough, the table of values confirms that as we approach 3 from below and from above,1037

the function itself approaches 9, approaches 9.1043

The table of values confirms what we thought.1050

The table values confirms our intuition, confirms our graphical intuition.1056

The limit from below, what we call the left hand limit.1071

I will often just call it 'lh'.1084

The limit from above, what we call the right hand limit, are converging to the same number.1090

That number is 9.1114

When the left hand limit and the right hand limit converge to the same number, we say that limit exists.1118

We say that the limit as x approaches 3 of x² exists.1131

We call this limit the number they converge to.1142

We call 9 the limit.1147

It is very important.1160

When you see a limit and it asks to specify whether it is a left hand limit or a right hand limit, you have to calculate both.1161

If the left hand limit and the right hand limit converge to the same number which they do, 9 and 9,1169

we call that number that they are converting to the limit.1177

We say the limit of x approaches 3 of x² = 9.1180

We actually write the limit as x approaches 3 of x² = 9.1187

That is our final statement, that the left and right hand limits are the same and they converge.1197

Let us see what we have got.1204

Very important idea, that is this symbol, the limit as x approaches a of f(x) does not say what happens when x = a.1206

It is asking you what is happening to f, as x gets close to a.1245

Not what is happening when x = a.1249

Distinguish between the two, that is probably going to be the most difficult thing, when you are starting out.1252

It does not say what happens when x = a.1259

It says what happens when x is near a.1267

y can appear to approach a value but that does not mean that,1285

Let me try this again.1318

y can appear to approach a value as x approaches a, just like we saw a moment ago.1325

As x approaches 3, it appears that y was approaching 9.1332

But that does not mean that f(x) is defined, it has to be defined.1338

It does not mean that f(x) has to be defined at a.1359

Now the previous example, it is defined at 3.1371

We know that 3² is 9.1374

There is some value 9 at when x = 3.1378

That is what the limit is asking.1384

The limit is asking what does it look like it is getting close to?1385

It looks like it is getting close to 9.1388

If I wanted to, I can take that 9 out and say the function is not defined there.1390

If I wanted to and I can do whatever I want with the functions.1396

The limits would still exist, the limit is still 9 from below and from above.1399

But at 3, the function does not exist.1404

We will see an example of that in just a moment.1407

y can appear to approach a value, as x approaches a.1413

But that does not mean that f(x) has to be defined at a.1417

Those are two independent things.1421

The limit of a number and the value of the function at the number are independent.1423

The limit as x approaches a of f(x) does not have to be f(a).1431

When that is the case, it is a special property, which we will talk about later called continuity.1449

In other words, it is a nice smooth curve, there are no gaps or breaks in it.1453

But it does not have to be that way.1457

Limit of f(x) does not have to be f(a).1468

It can be like the last example.1472

We had y which = f(x), which = x².1487

We said to that the limit as x approaches 3 of x² = 9, because the left hand limit and the right hand limit appear to approach 9.1492

In this particular case, f(3) which is equal to 3² = 9, they happen to correspond.1503

They do not have to correspond.1513

They happen to correspond, in this case.1520

They are actually independent.1530

The limit as x approaches a of f(x), sorry if I keep repeating myself, this was very important,1548

does not say, it does not say what is f(a).1557

If we want to know what f(a) is, we will ask you what is f(a).1568

This is asking you what is the limit as x gets close to a?1571

It says what happens to f(x) as x gets infinitely close to a, gets very close to a.1579

What is the behavior of f near a, not at a?1610

Let us do another example.1623

This one, I’m going to draw it out myself.1627

Another example, we notice that it looks like it is not defined.1633

I’m going to go ahead and draw a graph, because we want to develop some intuition.1643

Here is a graph, empty, and there.1648

We have a graph like this.1652

Let us say that this is 5 and let us say where that little point has been removed.1654

Let us say the y value is 7.1661

We ask the following.1664

This right here is our f(x), we ask what is the limit as x approaches 5 of f(x)?1673

It is not specified whether this is a left hand or right hand limit.1684

We have to approach 5 from below, the left hand limit.1687

We have to approach it from above to see what f(x) is getting close to.1690

We do the limit as x approaches 5 from below of f(x).1695

Let us see what happens as we approach 5, the function looks like it approaches 7.1702

The left hand limit is 7.1712

We will do the limit as x approaches 5 from above of f(x), what is that equal?1717

As we approach 5 from above, the function gets closer and closer.1725

We take the y values, it also looks like it is approaching 7.1731

The limit of f(x), since this corresponds, this is the left hand and the right hand, the limits are equal.1736

We say that the limit as x approaches 5 = 7.1742

The limit exists, we say that the limit exists and that this limit = 7.1751

Notice f(5) is not defined.1766

5, there is a hole here, it is not defined.1780

We do not know what it is.1784

The limit exists, the limit is 7 but f(5) does not exist.1786

They are very independent things and this is an example.1792

They are completely independent.1796

F(5) is not defined.1800

The limit as x approaches 5 of this particular function = 7 but f(5) does not exist.1803

'dne' means does not exist.1819

Let us see what we have got here.1828

Let us do another example.1832

I’m going to go ahead and draw this one out as well.1835

Another example.1839

Let us do this and let us do that.1844

This is our coordinate system.1846

We have some function like this.1849

This is some arbitrary function.1858

This is our f(x), and let us go ahead and say that this x value is 2.1860

Let us go ahead and say that this y value up here is 4.5.1867

Let us say that this y value down here is 1.5.1871

In this particular case, we ask what is the limit as x approaches 2 of f(x).1880

X approaches 2, let us not specify whether it is a left hand or right hand.1894

We have to do both.1897

The left hand limit is, when we approach 2 from the left, from below, let me go ahead and do this in blue.1899

When we approach 2 this way, the x values get closer and closer and closer to 2,1904

what does the function doing?1909

The function is getting close to 4.5.1910

The limit as x approaches 2 from below of f(x), it equals 4.5.1919

Let us do the limit from above.1930

We are going to approach 2 from above, from numbers that are bigger than 2.1931

We get closer and closer and closer and closer to 2, that means the function is going this way.1936

It looks like it is approaching the number 1.5.1942

The limit as x approaches 2 from above is equal to 1.5.1948

4.5 and 1.5 do not equal each other.1955

This limit does not exist.1958

The limit as x approaches 2 from below of f(x) which is 4.5, does not equal to the limit as x approaches 2 from above which = 1.5.1964

This means that the limit does not exist.1981

Notice f(2) does exist, the value of f(2) is that one right there, the solid dot.1986

It is actually 4.5.1993

The left hand limit exists, it is 4.5.1995

The right hand limit exists, it is 1.5.1999

But because the left hand and right hand limits are not equal, the limit does not exist.2002

We say that the limit does not exist.2007

Again, we can ask for a left hand, we can ask for a right hand, or we can ask for both simultaneously.2009

In order for the limit, when it is not specified to exist, the left and the right hand limits have to equal.2014

You see, you can have a left hand limit, you have a right hand limit.2022

You can have it be defined.2024

Three completely independent things.2025

Let us see what we have got here.2035

Let us do another example.2042

Let us try another example and I’m going to draw this one out as well.2046

This time we are going to go ahead and use the function 1/x, an actual function.2057

We know what this function looks like.2062

It is a hyperbola, it looks something like this.2064

We ask what is the limit as x approaches 0 of f(x)?2072

Let us see what happens.2088

As x approaches 0, here is our 0.2090

We need to the a left hand limit and we need to do a right hand limit.2095

We need to do, go ahead and do this in red.2098

The limit as x approaches 0 from below of 1/x, which is our function.2103

Let us see what happens as we approach 0 from below the function goes off to negative infinity.2110

That is what the symbol means, that is all it means.2122

It says as x gets close to some number, what does f(x) do?2126

It is very intuitive.2130

What is happening to the function?2133

We see what is happening to the function.2135

The function is just dropping down into negative infinity, that is the answer.2136

The limit as x approaches 0 from below, the 1/x is negative infinity.2140

Let us do the other one, let us approach 0 from above.2145

The limit as x approaches 0 from above of 1/x = positive infinity, because we see as we get close to 0,2151

the function, the function is going off to positive infinity.2160

Negative infinity and positive infinity are definitely not the same thing.2166

The limit does not exist.2169

If the function were different, if the function were both going like that, the left hand limit is going off to positive infinity.2173

The right hand limit is going off to positive infinity.2181

They are the same.2183

We say that the limit of the function is positive infinity.2184

Let me actually formalize what I just said.2201

We will do another example.2204

This time we will take the function f(x) is equal to 1/ x².2209

We know what that one looks like.2214

It is exactly what we just described.2217

There, and it is there, this are coordinate axis.2220

In this case, we want to know what the limit is as x approaches 0 of 1/ x².2227

What is it, we see as we approach 0, this is our 0, from the left, it goes to positive infinity.2235

As we approach it from the right, the function goes to positive infinity.2243

Therefore, this limit is equal to positive infinity.2247

It is very simple, graphs are really great.2252

They tell you exactly what is happening to a function.2258

A table of values gives you more information.2261

It actually gives you specific numbers, if the graph is not all that great.2264

Of course the last thing we are going to do, we are going to learn how to calculate limits analytically,2269

mathematically, to get a precise value for what it is.2273

You are going to use all three of these tools.2276

The graphical, the tabular, and the function itself, the calculus itself.2279

Let us take a look at this one.2287

This function right here.2298

This function is f(x) = 4x² + 2x - 5 divided by x² + x – 1.2299

I chose some random function that looks like, I wanted to be a rational function.2318

It would look something like this.2322

What we ask is the following.2325

This is our f(x), this is a graph of f(x).2327

Let me do it in blue actually, the little differentiation.2334

What is the limit as x approaches infinity of f(x).2339

What is the limit as x approaches negative infinity of f(x)?2350

That is it, we are just asking what happens when x gets really big, what does f do?2356

When x gets really big in the negative direction, what does f do?2362

That is all we are asking.2366

Let us take a look.2369

Based on the graph alone, let us do the first one.2370

We will do it over here.2374

The limit as x approaches positive infinity of f(x), it looks like as x gets really huge, it looks like the graph is approaching 4.2375

We are going to say it = 4.2392

It is approaching 4 from below.2394

Here, this limit, the limit as x goes to negative infinity, as it gets bigger in that direction,2398

the same thing, it looks like the function itself is dropping down.2407

It is getting close to 4.2411

It also equals 4.2415

In this particular case, when you are dealing with infinities, it is the same thing.2417

There are two basic conventions regarding infinity.2424

When you see the limit as x approaches infinity of f(x), some people take this to mean positive infinity.2426

They separate that from negative infinity.2436

Or when you see infinity, it means do both, do the positive and negative just like you would for x approaches 3.2439

You are going to have to do the x approaches 3 from below, x approaches 3 from above.2447

With infinities, we generally tend to keep them separate.2452

When you see the limit as x approaches infinity, it generally means positive infinity.2455

The limit when x approaches negative infinity, it is the negative infinity.2464

We definitely keep these separate.2468

Sometimes when I see the limit as x approaches infinity, I tend to just assume that it is both.2471

We are going to do both and we will specify which one we are doing,2477

when we are actually dealing with the specific problems.2480

In general, we handle the infinity separately.2482

Do a negative, do a positive.2485

They both happen to equal 4 but they are separate limits.2491

That is why we handle them separately.2495

You might have this limit be 5 and you might have this limit be 9, or it might be infinity itself.2499

The function might do something very different.2507

The right side of the graph, as opposed to the left side of the Cartesian coordinate system.2510

Just because the limit is 4 and the limit is 4, this is not the same as a left hand or a right hand limit.2515

Our left or a right hand limit is you are approaching a number from the left and from the right.2522

With infinities, they are separate because you are actually going to the right infinitely and to the left infinitely.2529

From your perspective, that is the right and that is the left.2537

They are separate limits.2539

We do not actually say because this is the case, we do not say the limit as x approaches infinity = 4.2542

The limit as x approaches positive infinity = 4 and the limit as x approaches negative infinity = 4.2549

They are separate limits, we do not combine these.2556

For infinite limits, again, treat them separately.2558

Let us look at another function.2566

This function right here is f(x) is equal to the sin of 5/x, that is what I have written here.2567

The question is, what is the limit as x approaches 0 of f(x).2583

Here is our 0 mark, we want to ask, we are going to approach 0 from the left,2594

we are going to approach 0 from the right because it was not specified.2601

You have to do both.2604

0 is a specific number, it is not an infinity.2606

This is a really wild function.2611

We see that it gets closer and closer and closer, the limit as x approaches 0 from below of f(x),2613

it just wildly jumping back and forth.2625

As we see that even if we move a little bit, like an infinitesimal amount, that function just jumps up and then jumps down.2629

It does not seem to be converging to anything.2636

This is -1 and this is +1.2640

It seems to be oscillating.2644

This is an example of a function that as you get closer and closer to a number,2646

the function itself starts bouncing back and fourth, oscillating between two numbers, +1 and -1.2649

It cannot decide, the limit does not exist.2656

In order for a limit to exist, it has to be a number and it has to get close to that number and stay close to that number.2660

The closer you get to a, that is the whole idea.2666

It converges, that word convergence in mathematics is huge.2670

It is everything in calculus, it is about convergence.2674

The same thing from the other side, when we approach 0 from the right, the same thing happens.2680

Here it is reasonable but as we get closer and closer to 0, it starts oscillating really crazy back and forth.2685

The limit as x approaches 0 from above also does not exist.2692

Here, the limit does not exist because it is oscillating between +1 and -1.2701

It is not converging to some single number or it is not going off to positive or negative infinity.2706

The idea of a limit, very important, that converges,2717

is that as x gets closer and closer to some a, that f(x),2736

the function itself, gets closer and closer to some number, to a finite number.2754

To an actual number that we can actually say 2, √6, 9, 4000, some finite number that we can point to.2767

Some finite number and stays close to that number.2775

Here it approaches 1, but then it jumps off to -1, then it jumps of to +1, it jumps back to -1.2785

It is no saying staying close to one of these.2791

There is no convergence, very wild function.2793

Let us see what else we have.2800

Let us go ahead and round this out.2802

Let me go back to blue here. We have seen the following.2807

We have seen the limit as x approaches a of some f(x) = l, some finite number, some finite actual number.2816

That was one thing that we have seen.2836

We saw the limit as x approaches a of f(x).2838

We see it go off to positive or negative infinity, like the function for 1/x.2845

The a was 0, it is approaching some specific number but the function itself is flying off to positive or negative infinity.2850

We also saw an example of the limit as x itself approaches infinity of f(x) = l, some actual number, that was the rational function.2858

We saw that as x gets really big positive, really big negative, the function itself got close to 4.2876

4 is an actual number.2886

This time x was approaching infinity.2887

Maybe from pre-calculus you remember, any time we take x to be going positive or negative infinity, we called it end behavior.2893

The limit as x approaches positive infinity of f(x) is asking what is the end behavior of the function.2902

We also saw that as an example of the limit as x approaching a from below, not equaling the limit as x approaches a from above.2908

Here the limit did not exist.2921

In order for a limit to exist, an actual finite number or positive or negative infinity,2929

the left hand limit and the right hand limit, as x gets close to a single number, f has to go to the same number.2935

They have to equal each other.2943

If they are not equal to each other, the limit does not exist.2945

The left hand limit exists, the right hand limit exists, but the limit itself does not exist.2948

All things are possible, you might have a left hand limit exist but the right hand limit does not exist.2956

Anything is possible.2959

One more time, I know you are going to get sick and tired of hearing it.2964

I’m certainly sick and tired of hearing myself saying, but it is very important repetition.2968

f(x), the limit, the symbol, limit as x approaches a of f(x) is asking, as x gets arbitrarily close to a, what is f(x) doing?2971

That is it, very intuitive, use your intuition.3003

You have the graph, you have the table of values, and you are going to learn to do this analytically.3008

It either converges to a number, diverges to positive or negative infinity, or oscillates, or it just not does not exist.3013

Those are the possibilities.3048

Oscillates is the same thing.3049

When something oscillates, it does not exist.3051

It either converges to a number, it exists.3054

It diverges to infinity, positive or negative, or it does not exist.3056

Those are the possibilities for a limit and that is all.3061

Let us round it out.3068

When the limit as x approaches a of f(x) = l and the limit from below,3072

the limit as x approaches a from above of f(x) = l, we say that the limit exists.3082

The limit as x approaches a of f(x) = l.3102

There you go, thank you so much for joining us here at www.educator.com.3109

We will see you next time, bye.3113