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Calculus: Limits and Indeterminate Forms


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About this Lesson

  • Type: Video Tutorial
  • Length: 18:55
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 204 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

Calculus (279 lessons, $198.00)
Calculus Review (48 lessons, $95.04)
Calculus: Final Exam Test Prep and Review (45 lessons, $64.35)
Calculus: Limits (12 lessons, $19.80)
Calculus: Evaluating Limits (4 lessons, $8.91)

The limit is the range value that a function is tending towards as you get closer to a domain value. In this lesson, you will learn how to evaluate a limit at a value where a function is well-behaved by substituting the value into the expression. If direct substitution produces zero divided by a non-zero number, the limit is zero. If direct substitution produces a non-zero number divided by zero, the limit does not exist. You will also cover how to handle limits that produce indeterminate forms (0/0) that may or may not exist. An indeterminate form, unfortunately, is just a sign that more work should be done and Professor Burger walks you through what should be done in what order to determine whether there is a limit (and what it is).

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/calculus. The full course covers limits, derivatives, implicit differentiation, integration or antidifferentiation, L'Hôpital's Rule, functions and their inverses, improper integrals, integral calculus, differential calculus, sequences, series, differential equations, parametric equations, polar coordinates, vector calculus and a variety of other AP Calculus, College Calculus and Calculus II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

~ joel8


I love this guy
~ jfilip

This guy is a great teacher. The limit problems he is doing once made me cringe but they seem much more manageable now. Still a little scary, but I know how to tackle them now. Great lesson.

~ joel8


I love this guy
~ jfilip

This guy is a great teacher. The limit problems he is doing once made me cringe but they seem much more manageable now. Still a little scary, but I know how to tackle them now. Great lesson.

Evaluating Limits
Limits and Indeterminate Forms Page [1 of 1]
The only way to make something your own is to do it. Just to get in there and just do it. So what I’m thinking about now is just a marathon session of doing lots and lots of examples and to really get a feel for this notion of coming together mathematically, the notion of a limit. Okay and the more you do it, the more you’re going to get comfortable with it and the more you’re going to make it your own. And really, what I invite you to do is to work through these with me—in fact, it’s going to be very interactive, I’ll pose a question and then actually ask you to try it on your own. Try it, do as much as you can, and then, try to make a guess—try to make an answer—come up with an answer if you can, and then keep going and I’ll do it for you. And then after you do this and do the examples that we have for you, I suggest you actually go and crack your Calculus book open, do every single limit problem in there. Sounds extreme. There’s, like, sixty of them in there. I’m telling you, folks, you do sixty limit problems, you’re going to know a limit until you go to the grave, seriously. So that’s the way to get it down. Just a lot of them and then you really understand them.
Okay, let’s begin with the first one. Let’s find the limit as x approaches minus one of x cubed plus the square root of x plus five. I would like for you to try this right now, in fact. So, give it a shot. We’re going to stop for a second. Once you’ve got it, just click on the key, and we’ll keep going.
Have your guess? All right. Let’s see what I guessed. Well, the very first thing I always do is I ask myself, “What is going on at this function at the exact point of minus one?” Many of you are thinking, “Gee, square root minus one, square root minus one, sounds a little scary.” Well, maybe it is, but I’m not afraid to try. So if I plug in a minus one here for the x cubed, I would see a minus one cubed plus, and then I would see the square root of minus one plus five. Well, minus one plus five actually turns out to be four. Square root of four is actually a very happy thing. It’s just two. So this is two here, and minus one cubed—minus one times minus one times minus one—is just minus one. So actually, what I see here is minus one plus two. So, that equals one. I get an answer. I get a number. It looks great. Turns out that therefore, this limit exists and equals one. Not too hard. Again, first thing I always try is to plug in things. If the plug-in thing succeeds, I’m happy. If the plug-in thing fails, problem’s a little more interesting.
Let’s try this one. Limit as x approaches one of three x squared minus two x minus one, all divided by x over one. Where do I come up with these problems? I’m not telling all my secrets. Anyway, give this one a shot. See if you can come up with a guess, and we’ll see if it’s right.
Well, maybe you guessed that, in fact, the limit is undefined—the limit doesn’t exist. And that would be a great guess if you made that. Let’s see if that’s right. So what I would do here first is I would plug in the one. Wherever I see an x, I’m going to plug in one. So, let’s see, if I plug in the one here, I see three times one squared, which is three, minus two times one. So that’s going to be three minus two, well three minus two is one, one minus one is zero. Okay? And in the bottom when I plug in one, I see one minus one, which is zero. So I see zero over zero, and so this limit is undefined—doesn’t exist. Did you guess that? If you guessed that, then you made one of my favorite mistakes in this business. It’s the one I warned you about in our previous discussion. Do you remember that? I said if you get a limit of zero over zero, you’ll be so tempted to say it’s undefined, but remember, it’s not undefined, it is an indeterminate form. It’s an indeterminate form.
So, if I fooled you—and in fact, there might be some people, by the way—let me just tell you this little thing. There might some of you that said, “Gee, I got an answer, but Burger is now saying that it doesn’t exist, so I must have made a mistake.” Now, if you thought to yourself—if you actually doubted yourself—if you got answer you doubted yourself, because I said it doesn’t exist. Then, I just want you to do something really painful to yourself. I don’t know what. You know—just, just—I don’t know—hit yourself on the hand or something like that. Because you should have more faith in yourself, and you shouldn’t just trust me because I say something. That doesn’t mean it’s true. You’ve got to convince yourself that it’s true. And if you have an answer, stick by it. Turns out it’s wrong, who cares? But stick by your answer and be confident. And when someone says the answer is wrong that you have, force them to explain why it’s wrong. Just don’t believe them on faith. Okay?
If you didn’t get—uh—if your answer wasn’t “it doesn’t exist,” good for you. Let’s see if we can figure out what the actual answer is. Remember I plugged in and got zero over zero. That means that this needs more work. Okay, well what more work can I do? Well, a nice idea is to always ask; “Can I factor? Can I isolate that zero over zero?” as we saw in the previous example. So, well, can I factor the top? Well, that might be something we could attempt to do here. Let me try to factor the top. Again, I remind you that a lot of people would just start writing the function here and doing stuff. Please, please, always carry that limit, because it’s reminding you that you want to take a limit later in life. And now, can we factor the top? Well, I think we can, and here’s how we start. I just make that _____ therefore it can be factored. See, if you can do that, you can factor it. Just a joke.
And now, what you want to think about is how to factor it. Now, in fact, if you really become a little bit savvy here, by the way, let me just tell you something that no one else would admit to you here. But, you know, if you’re trying these questions, and you’re trying to figure out how to factor this—well, if there’s going to be a zero over zero, maybe x minus one will be one of the factors. That actually might be not a bad hint. But, we can’t use that for sure unless we see its right. But it’s a good foreshadowing; maybe one of these will be the x minus one, so we can cancel them, but who knows. I’m just talking to you as the savvy, web user.
Well, let’s see if we can factor this. I’m going to put a three x here and an x here. This tells me that I need a plus or a minus, so I could either have a plus here or a minus here or a minus here and a plus here or a minus here and plus here—all these different ways. And, let’s see, I bet you a minus here is going to work and a plus here. And I bet you a one and a one’s going to work. And, sure enough, that’s one x minus three x is a minus two x and this gives me the minus one. Voila! I factored and, more importantly, that little foreshadowing thing turns out to actually be not so crazy. I see a factor of x minus one on the top and a factor of x minus one on the bottom.
So, I could happily cancel those things away, and I’m going to cancel them right away with my cancellation pen, right here. Cancel those guys away. Only one thing you’ve got to promise me. What do you have to promise me? You have to promise me that the thing that I’m canceling away is not going to equal zero. You have to promise me that x minus one does not equal zero. Which means what does x not equal? X does not equal one; X does not equal one. Do you have a problem with that? No, because we’re just approaching one. The beauty of the limit. We’re looking at this function through a window that has a frame. So what happens to that one is of no interest of us. All we care about is what happens close, but not equal to one. Cancel away. Absolutely no problem. Okay? Well, what are we left with? Well, then this equals the limit as x approaches one of just that term, which is three x plus one. Well, and that’s a line. We can actually figure out what that limit is. We plug in the one in here, and we see three plus one is four. This answer is four. The limit is four.
Again, a little bit of work involved in this. A lot of algebraic, sort of abstract, steps here. How can we sort of visualize what’s going on? Well, it turns out, this is another example of something that we’ve already seen earlier, again in the parabola section of graphing. We actually graphed this thing and thought it was just this line. In fact, take a look at the graph right now. It’s over there. It’s just this line with a hole at x equals one. And while this question is asking us, “Where is that graph—uh, what value is that graph approaching as the x’s get closer and closer to one?” And if you look at the picture, you can see that if you run along that line, your fingers will be coming together. They’d be approaching that hole. So, I admit there’s a hole there, but your fingers do want to come together. And the question is what value would your fingers touch at? Your fingers would touch—take a look at the picture—your fingers would touch where? Well, at four, that’s what we got here.
So, there you see it graphically. Here you see it algebraically. Of course, the thing is, if I put a really complicated problem here, the algebraic moves will still work, whereas it’ll be harder to look at the graphs of these things. But we should get a flavor for what’s going on in the problems that aren’t so painful.
Okay. Well, now, let’s have you try another one of these. See how you’re making out with these things. Once you get into the rhythm of them, not so bad, not so bad. They look pretty horrendous, by the way, but you know, looks are deceiving.
How about the limit as x approaches five of this—uh—x squared minus four x minus five all divided by x squared minus six plus five. Okay, so we have x squared minus four x minus five divided by x squared minus six x plus five. Why don’t you take the limit as x approaches five? There’s a lot of five’s in this problem, folks. I want you to try it right now. Try it, and we’ll work it out together.
Have your answer? Let’s see how we fared. First thing I would do always is I plug in the five and see what the function reveals. Maybe the function’s actually going to be quite nice around five. I don’t have to do any work. A lot of people, by the way—here’s a classic little blunder—not a mistake, just a little blunder. People look at these problems, they immediately think, “I’ve got to factor! I’ve got to factor!” And they do all this factoring work. Turns out, this wasn’t a point of contention. This wasn’t a problem. They could have just plugged in, got the answer, no problem.
So that’s sort of a sneaky way that profs can sort of make you do—jump through all the hoops—without actually needing to jump through the hoops. Only jump through a hoop when you got to. That’s what I say, and also, make sure the hoop’s not too high, because otherwise, you could hurt yourself.
Okay, now, how would you go about this? First, I plug in the five. So let’s do that together. I plug in the five here, and I get five squared, which is 25, minus four times five, that’s 20. So that’s 25 minus 20, which is five. Five minus five is zero, zero on the top. Zero on the top is not going to scare me, because that could be potentially zero if I got a four on the bottom. Zero over four would be zero. What I got on the bottom though—well, I got another 25, 25 minus—in this case—minus 30, 25 minus 30 is minus five, plus five. Minus five plus five—zero; zero over zero—indeterminate form. Doesn’t exist? I don’t know. Need’s more work. If you get nothing else out of this little lecture—this little discussion—it is that when you get zero over zero in a limit, it’s an indeterminate form. It needs more work.
Okay. Well, what kind of work can we do? There’s only one technique we’ve seen so far, and it’s a good one—see if we can factor this thing. So, let’s try to factor and see if we can make this thing look nicer. Again, I write the limit. I told you, I’d like for you to write the limit all the time. You can write it very sloppy. You can write it very small. Any way you want. Any font you want, but always write it. Let’s see if we can factor the top here. Let’s try this on the fly, again. There, I put an x here and an x here. I see a minus sign here, so I’m going to need two different signs here. I’ll put a plus here and a minus here. I need two numbers that multiply to get five and then combine to get a minus four. So I would think a five and a one might serve me pretty well. That’s a minus five x plus one x is minus four. Perfect. I hit that one right on the target. And what about factoring the bottom? Let’s try to factor the bottom together. I see the x squared so I put in an x and an x. The plus signs tell me both of these are going to be the same sign, and that same sign is that. So it’s a minus and a minus. Is this how you learned it, by the way, wouldn’t if be funny if the way they taught it back in the stone age, when I took it, is the same way they’re teaching these days? I don’t know. And let’s put in something that’s going to actually give five. So how about a five and a one again. And then we see a minus five x minus one x is a minus six x. I factored this thing nicely. Great.
Now look at that. Do you see the problem? Do you see the zero over zero of this? Take a look at this. Find the zero over zero. Do you see it? It’s right here. Here’s the zero over zero. Do you see that term? And, happily, I can cancel those things away. If I cancel those things away—cancel is always a proviso. There’s always a cost to canceling. You can’t cancel zero. So x minus five cannot equal zero, which is another way saying x cannot equal five. X cannot equal five. Does that worry you? Does that keep you up at night? No, it doesn’t worry you. It doesn’t keep you up at night, because I taking the limit as x approaches five. Taking the limit as x approaches five, it’s completely okay to have x not equal to five. All I’m doing is approaching five. I don’t care what happens with five.
Do you see the beauty of limit? See the beauty of blocking off that one point. It allows you—empowers you—to do things that you otherwise couldn’t do. I don’t care what’s going on at five, only very, very nearby, and nearby is not equal to five. So this happily reduces to the limit as x approaches five of—and what’s left over? Well, we’ve got this x plus one on the top and on the bottom, we have x minus one. Now a great mistake would be to actually forget to write the bottom. Write the bottom in. X minus one.
Okay, well, now what’s this limit. It still looks like a fraction. Still looks a little scary, but the first thing I always do when I look at a limit problem—even if it’s in the middle of another limit problem—is to ask, “Can I just solve it easily?” So, let’s insert the five and see what we get. By putting the five here, I see a five plus one, which is six. If I put a five in here, I see five minus one, which is four. That’s not zero over zero.
By the way, notice that I don’t actually write those things out over here and then discover zero over zero. I actually either do it on the side or I do it in my head, because you don’t want to write equals zero over zero. You want to check. If you get zero over zero—indeterminate form—needs more work. If you don’t get zero over zero, then you actually produced an answer.
So here, in this case, I see I’m not getting zero over zero, so I can write in six divided by four, which can reduce to three over two. So this limit actually exists and equals the number three over two. Hope you got that one. And if not, I hope you see now how to look at this.
Let’s try two very quick, short ones, and then we’ll wrap things up here. So the first one I want you take a look at is the following. Let’s take a look at the limit as x approaches three of x squared minus x minus six, all divided by x squared plus three x plus two. So why don’t you give this one a try right now, see if you can come up with a guess.
Well, let’s take a look and see what we’ve got going on here. If I insert the three in here—the first thing I’m going to do again is I’m going to insert that three wherever I see an x. A lot of people start to factor, like said, I wouldn’t do it. First, make sure the easy way doesn’t produce an answer. So you’re plugging a three in here. What do I see? I see a nine minus three. So nine minus three is six, six minus six is zero. So I get a zero on the top, okay? Let’s see what’s going on in the bottom. Shall we take a look under the hood? I see a three squared which is nine, plus a three times three, which is another nine. So nine plus nine is 18, 18 plus two is 20. So I see zero over 20. Is zero over 20 an indeterminate form? No. Indeterminate form—zero over zero. Zero over 20—undefined? No. Zero over 20 is the number zero. It equals zero. Zero over any non-zero number is zero. So this limit is zero.
Okay. One last one, and then we’re going to call it close here. Limit as x approaches minus one of x plus five divided by x squared minus one. Limit as x approaches minus one x plus five divided by x squared minus one. Give this one a try.
What did you get? Try any factoring stuff? Factoring is a great way to start, but always—first thing I do is plug in and see what I get. Plug in a minus one on the top and I see a minus one plus five is minus four. Minus four, right there I know I’m not going to get an indeterminate form. I’m not going to get an indeterminate form. So, what am I going to get? Well, I don’t know, but it’s not going to be indeterminate. Indeterminate is zero over zero. When I plug in the minus one downstairs what happens. I see a minus one squared. Minus one squared is actually one. One minus one is zero. So, now I see four divided by zero. Is four divided by zero—an indeterminate form? No. It is not an indeterminate form. Four over zero is something that is completely understood. It is undefined. You can’t take a non-zero number and divide it by zero. There’s no ambiguity as with zero over zero. Can you cancel? Can you not cancel? What’s the story? Is it one? Is it zero? Is it undefined? But four over zero, like seven over zero, or minus three over zero, in undefined. This limit does not exist. This limit does not exist.
So this is the first example we’ve seen where a limit, in fact, does not exist. And how did we get that? Because we got a non-zero number divided by zero. That means that we—our fingers are not coming together. Remember for a limit to exist, those fingers have to be heading toward each other. Turns out that in fact, with this particular function, our fingers are not heading toward each other. They’re going far, far away. They’re not approaching a particular number, and that’s why this limit doesn’t exist. So, remember zero over zero—indeterminate form—needs more work. If you get zero over anything that’s not zero, the answer is zero. If you get anything that’s not zero divided by zero, the answer is limit doesn’t exist, undefined.
Okay. Hey, you have some fun with these things, and when we come back next, what we’re going to take a look at are some more interesting examples where, in fact, this factoring idea may not work. What could those things even look like? Stay tuned, we’ll see you in a second. See you in a bit.

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