For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

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### The Limit of a Function

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- The Limit of a Function 0:14
- The Limit of a Function
- Graph: Limit of a Function
- Table of Values
- lim x→a f(x) Does not Say What Happens When x = a
- Example I: f(x) = x² 24:34
- Example II: f(x) = 7 27:05
- Example III: f(x) = 4.5 30:33
- Example IV: f(x) = 1/x 34:03
- Example V: f(x) = 1/x² 36:43
- The Limit of a Function, Cont. 38:16
- Infinity and Negative Infinity
- Does Not Exist
- Summary 46:48

### AP Calculus AB Online Prep Course

### Transcription: 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 analysis *0614

*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

1 answer

Last reply by: Professor Hovasapian

Wed May 11, 2016 3:40 AM

Post by Tom Edison on May 9, 2016

Hi professor Hovasapian.

How would you explain f(x)=1/-x

Thanks.

Your pupil