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Calculus: Two Techniques for Evaluating Limits

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About this Lesson

  • Type: Video Tutorial
  • Length: 17:56
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 193 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. This lesson will cover the cancellation technique and the rationalization technique to evaluate and solve for limits which give you an indeterminate form when evaluated with substitution. When evaluating the limit of a compound fraction, Professor Burger will teach you to simplify the fraction by using the lowest common denominator (LCD). He will also walk you through how to simplify a binomial that contains radicals by rationalizing (multiplying by the conjugate of the binomial to remove the radical). The conjugate of binomial a-b is the binomial a+b.

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

Thinkwell
Thinkwell
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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...

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Recent Reviews

Nopic_blu
Limits with square roots and unfactorable ratio...
09/08/2012
~ Jodi7

Very helpful!
It's been over ten years since I sat in Calculus and now I have been thrown last minute into teaching it. These videos are exactly what I need to teach well and have my students learn.

Nopic_grn
AAAAAAA+++++++++
04/23/2010
~ joel8

He is ALWAYS GREAT!!!! MUST WATCH!! THANK YOU!!!! Hopefully people can truly appreciate how rare it is to have the ability to watch one of the best professors utilize all skills of teaching to produce such power packed lessons that stick in the brain. Every detail is thought of in the video presentation. THANKS AGAIN!

Nopic_tan
This totally cleared up any confusion I had wit...
07/17/2009
~ brittanie

Undefined limits are awful! I'm so glad there is an online video tutorial explaining this to me in baby steps. Very thorough instructions using two examples on how to find the limit of a square root function. Totally happy with my purchase. Thanks!

Nopic_blu
Limits with square roots and unfactorable ratio...
09/08/2012
~ Jodi7

Very helpful!
It's been over ten years since I sat in Calculus and now I have been thrown last minute into teaching it. These videos are exactly what I need to teach well and have my students learn.

Nopic_grn
AAAAAAA+++++++++
04/23/2010
~ joel8

He is ALWAYS GREAT!!!! MUST WATCH!! THANK YOU!!!! Hopefully people can truly appreciate how rare it is to have the ability to watch one of the best professors utilize all skills of teaching to produce such power packed lessons that stick in the brain. Every detail is thought of in the video presentation. THANKS AGAIN!

Nopic_tan
This totally cleared up any confusion I had wit...
07/17/2009
~ brittanie

Undefined limits are awful! I'm so glad there is an online video tutorial explaining this to me in baby steps. Very thorough instructions using two examples on how to find the limit of a square root function. Totally happy with my purchase. Thanks!

Limits
Evaluating Limits
Two Techniques for Evaluating Limits Page [1 of 1]
The cool side of limits – okay, in all the examples of limits we’ve seen so far, they fall into sort of one of two categories. Either we can immediately figure out what the answer is by plugging in and seeing we actually get the answer or seeing that the limit might not exist. Or the other possibility is we get zero over zero, which is an indeterminate form, and it needs more work. And the technique we’ve seen so far for conquering those zero over zero indeterminate forms has been to factor. It’s a powerful technique; it’s worth a lot.
Okay. So, it turns out that there are examples of problems where the factoring technique—the factoring trick—is not going to cut it. And I wanted to show you two such examples right now of those kind of things. These examples actually are really important for what’s going to happen in terms of our goal—our inevitable goal of finding instantaneous rates of change, instantaneous velocities. And these are actually important examples, not just weird ones. Let me give them to you, and then I want you to try to actually resolve them. I know that I haven’t told you what the trick is yet. I haven’t told you how to do these problems, but why not give you a shot to at least try. I think you may be able to sort of fool around and at least try some false starts if nothing else.
Here’s the first one. Looks pretty threatening. The question is can you find the limit as x approaches zero of one over x plus one minus one, all divided by x. Sort of a complicated-looking thing. The limit as x approaches zero of one over x plus one minus the number one, all divided by x. Take a few seconds now and try this one on your own.
Give it a try? Any progress at all? If you didn’t make any progress, terrific. The important thing here is to just try and to come up with something. I didn’t tell you how I would tackle a problem like this, but maybe you figured out a way of thinking about it. First thing I would try, by the way, is the old favorite. I would just plug in x equals—in this case—zero, and see if I can do it. Well, what happens? Let’s do that together.
Plug in x equals zero; I see one over zero plus one. One over zero plus one, that’s just one. One minus one is zero on the top. Zero on the bottom. Indeterminate form. Needs more work. Let’s factor. Well, how are you going to factor? There’s no squares. There’s no cubes. I can’t factor out an x. So, actually, factoring seems to actually not be such a hot idea for this problem. There’s really nothing that I can really see to factor. So, what should we do now? Well, the answer is, “I don’t know.”
So we need some sort of new idea. Well, let’s think about this. If we look at the problem—and you know what? If you look at the problem and listen really closely, you can hear that the problem is actually telling us what to do. It’s crying out. If you listen really closely, you can hear it, just listen. Do you hear it? Look at this problem. What makes this thing so weird-looking? What makes this thing so bizarre is there’s fractions within fractions. Fractions within fractions. It’s obvious. We have to clean this thing up. It’s just so complicated. Right? Figure out what you don’t like about it, and then get a way of untangling the very thing you don’t like.
In this case, it’s all this fraction stuff. So what am I going to do? I’m going to try to get a common denominator here, and actually combine these two things and make it into one big fraction, and then take that fraction with this and make it into one huge fraction. So whenever you see fractions within fractions, a good technique is just to combine them and try to clean things up. Let’s do that together and see what happens. Well, what does this equal? It equals—again—the limit. Remember, I always write limit as x approaches zero. And now, I’m going to get a common denominator here. Now, the denominator here is x plus one. That’s actually a fraction, though it may not look like a fraction to you. It’s actually one divided by one. One divided by one. So, to get a common denominator, I multiply top and bottom here by x plus one. So I’ll see x plus one over x plus one. I’m going to write that out, actually, because there’s a lot of steps here.
Now, you know these problems start to get really sort of long and involved. But the pay off is that it’s sort of fun at the end when you actually get the answer. You feel like you just climbed Mount Everest—you know—it’s sort of neat.
I just got a common denominator—notice—that really does equal the previous line, because that’s just the number one, again. But now, I’ve got a common bottom, so I can actually combine these things. Remember how you combine. You subtract. So, I would get the limit – again, it’s a very, very narrow font, but it’s still there, and that’s all that matters.
And then, what do I have? I have one minus x plus one, all divided by the common bottom, x plus one, and then that’s all divided by this x. Now, I want you take a look at that. I want you to sign off on this. I want you to say that, “Yep, this is okay with me. I got the common bottom, and I’m all set”—subtracted correctly. Take a look at that for a second, and I want you to make sure that you’re with me on this one. You with me?
Now, I want you to tell me why this is wrong. Why is it wrong? Now, you’re saying, “Wait a minute. I thought this was right. You wrote it down.” Don’t believe me because I write something down. Don’t believe anyone because they tell it to you. You’ve got to convince yourself of this. This is wrong. And, in fact, I think I made the classic Calculus and above math student mistake. And that is, this fraction here, you know, the entire top wanted to be subtracted—the whole top. It was a team effort. And what did I do? I just subtracted off the first person, and I cut the second player. Well, that wasn’t great. I’ve got to subtract off everybody. So, the mistake I made, which is a classic mistake—hope I fooled you—really hope I did, because then, I hope this will teach you a lesson, and you will never make this mistake later in life—and that is, if you’re subtracting a whole bunch of stuff, you’ve got to put these parentheses. You’ve got to put these parentheses around it. I’ve got to subtract everything. That negative sign—I’ve got to distribute that thing. I’m going to share that with everybody. I’m going to share it with the x. I’m going to share it with that one. Really important, and really important, in particular, with this problem. Okay, you see that mistake? Great.
All right. Now, what do I do next? Well, now I say this equals limit as x approaches zero. And now if I distribute that, what do I see? Well, I see a one minus x distributing minus one. So I see one minus x minus one. Well, one and the minus one, they cancel. I’m just left with a negative x on top. So on the top; I’m just left with negative x. And on the bottom of that, I’m left with x plus one, and then I’ve got a real big bottom, and I’ve got the x there.
Now, what do I do? Well, again, I think of this x as, actually, x over one, and I invert the complex fraction. So I invert—whoop—and I multiply. When I do that, I see limit as x approaches zero of minus x over x plus one, and then when I invert that, I see times one over x. Okay?
Look! Can you see the zero over zero? The zero over zero is right there. Right? X is going to zero. So this is a zero over zero. So, actually, this work has given me the zero over zero term. You cancel that away. It’s a factor. I can just cancel that away, as long you make me a promise—you already know about this now—x cannot be zero. Is x zero? No. X is just approaching zero. It doesn’t equal zero. This is completely okay. And now, what am I left with? Well, now, I’m left with the limit as x approaches zero of what? Well, now don’t forget that lonely negative sign, right there. There’s a negative sign there. Don’t cancel that away with all that cancellation. It’s still there, and it hits that one and makes it a negative one. And you divide that by this term, which is x plus one. And what’s that limit? That limit we can just do by the easy method of plugging in. Let x equal zero. That term goes away down there. I’m left with negative one over one, which equals negative one. Negative one.
So this limit, which looked pretty mysterious, turns out to exist and equal negative one. Whenever you see fractions within a fraction, the trick there, just make it into one big fraction. Things will drop out, and it’ll be great. Okay? That’s sort of fun.
Now let me show you one other example where things sometimes get into a little bit of a messy state, but we can easily rectify. This one is the limit—here’s the question—the limit as x approaches zero of—now brace yourselves—okay, square root x plus four minus two all over x. I’m sorry about this, but it has to be said. It has to be said.
All right. In fact, I’m not even going to let you try this one. Don’t worry about it. Let me do this one for you, because this really is one that, you know, you shouldn’t be doing on your own until you have been equipped with the knowledge here.
First thing I would do is let x go to zero and see what happens. If we plug in zero, I see the square root of four. The square root of four is two, but two minus two is zero. So you get zero on the top and zero on the bottom – indeterminate form. Okay, so this needs more work. Can you factor? Well, not with that square root thing there. You can’t factor square roots too easily. Can we try to combine the fractions? Well, there’s not a lot of fractions there to combine. It’s just the square root thing.
Here is the trick. It is a trick. I am telling as my new virtual friend, this is a trick. Should you accomplish this on your own? No. How did anyone else come up with this? Well, they tried doing these problems a lot, and they came up with this trick. The trick is—in mathematics, if something gets really, really hard, there’s one of two things to try. Add zero or multiply by one. Notice that if you add zero to something, it doesn’t change, and if you multiply it by one, it doesn’t change. But these two tricks, actually get you through some of mathematics thickest and toughest problems. And here’s a wonderful example whereby multiplying by the number one, we can actually turn this impossible-looking problem into a resolvable problem.
Now, the trick always—whether you add zero or multiply by one—is coming up with a very clever way of writing one. Now what in the world does he mean by, “coming up with a clever way of writing one? The only way I know to write one is like this.” So, what do I mean? Well, I’ll tell you exactly what I mean. Remember that if you take any number that’s not zero and divide it by itself, that is the number one. So what I’m going to do is I’m going to multiply this by the number one. And the way I’m going to write the number one is I’m going to write something, and I’m going to divide it by itself so they cancel out and just produce a one. And one times anything doesn’t change the value, so that equal sign is completely legal. It’s going to be the same thing.
Now, what’s the one I’m going to choose? Well, what’s the problem with this thing? What’s the thing that you hate about this problem? It’s that darn square root. You hate the square root. I know you hate the square root. So, we want to get rid of the square root. Now a wonderful guess, a terrific guess, a spectacular guess would be to write in—well, you know that if you multiply a square root by itself, the radical lifts. So this, I think, would be a wonderful guess. And I hope you maybe have even thought about this on your own. Because then, when you multiply these two things out, you see, it’ll lift the radical. The radical goes away.
So that’s a great guess. You should try it and see what happens. It turns out it’s actually not going to work, because, in fact, when you do that, you’re going to get a whole bunch of extra radical stuff on the top that’ll be a little different than the first radical you had. But it turns out it’s a great idea—it’s a great idea—that won’t work. People think that if an idea doesn’t work, it’s not a great idea. That is complete bunk. You can have a great idea that doesn’t work. It’s okay. Because, in fact, this idea can be modified very easily to become an idea that doesn’t work.
Instead of putting in the minus sign here just like it was here, let’s actually change this and make it a plus sign. So, actually, this has a fancy name. This thing here is called the conjugate of that thing. See, I’m going to write it for you there. Conjugate. The conjugate is just—when you have a thing with a square root—when you rewrite it, but change the sign here, that’s called the conjugate. And notice I multiply top and bottom by the conjugate.
This is the trick. The trick, folks, is when you see—when you see a square root like this, multiply top and bottom by the exact same objects, but change the sign in between them. Okay, let’s see how that works. Again, you may be thinking to yourself, “I would have never come up with that.” Or you may be thinking to yourself, “Why should I do that?” Both of those two things are natural reactions. “I could have never come up with it.” You’re absolutely right. Neither could I. And “Why does it work?” You’re going to see why in a second.
Let me point out that if you were to work on these kind of problems for a long, long time and not just for a couple hours, you would actually would have come up with it. But, I’m trying to share with you knowledge that people already have.
So, to multiply these two fractions, I multiply the top and I multiply the bottom. Now, in these problems, by the way, there’s one great thing you should do. Never multiply the bottoms out. Keep the bottoms just as they are. So x times square root x plus four plus the number two. See. There’s a times sign there. Are you happy with that? No, you are not, because that x has a desire to be intimate with every single thing here. So, we have to make sure that that thing actually has an opportunity to socialize with everybody. I put parentheses around that.
But don’t distribute the x. Keep the x separated from these people, all right? It might be—it might be infected. So, just keep that out, and then keep everything just like that—factored form on the bottom. Always keep the bottom factored. It’s a great way to save you some time. The top, though, we have to expand out. Let’s expand the top out together. So, I’m going to FOIL this. The first times the first. So that’s the square root multiplied by itself so the radical just lifts. So the radical lifts, and I’m left with x plus four.
Now you’re going to see why it’s so important to put this difference in signs here. Because notice my inside term is negative two times the square root, but my outside term is positive two times the square root. Negative two times that square root, and then I add positive two times the same square root. What do you notice? Those terms cancel. That’s the power of switching those signs. You get the cancellation and no square roots. Look, Mom, no square roots! Okay, and then—so that drops out, and then the last times the last is a minus four.
Look how nicely that cleans up. Now, you may be saying, “Well, gee, I just traded in, you know, one hard problem for another hard problem. There were square roots here, now there’s square roots no longer on the top, but now they’re on the bottom.” Well, I admit, you’re absolutely right. The square roots are no longer on the top; they are now on the bottom. But you’re wrong when you think you’ve traded in a hard problem for another hard problem. You just traded in a hard problem for an easy problem. Because you’ll notice that the four minus the four cancel right there, and then, what do you have? You just have x on the top, and what do you see? I’ve once again triumphed over this problem, because I’ve isolated the zero over zero. Do you see it? X over x times something. So, I can now cancel those things away. I can cancel those things away. That’s why it’s good, by the way, not to distribute that. Do you see why? It’s good to keep it factored out, as long as I abide by the rule x doesn’t equal zero. Does x equal zero? No! X is only approaching zero. So this is completely okay. X is never going to equal zero in this problem. And now I’m left with the limit as x approaches zero—and notice I keep writing that.
What’s on the top? Sort of messy, but you’ll notice nothing’s on the top. Should I write zero on the top? No. I’ve cancelled everything away, leaving me with an invisible one. Don’t forget, there’s a one multiple on top here, not a zero. I’ve cancelled things away and replaced them by one’s. And the bottom, this is gone, but that still remains, so let me write all that in. And don’t let that square root, by the way, scare you. In fact, you know, none of these things should scare you. Like, you know, students get scared by this. I want to show you something. This cannot hurt you, and let me just prove it to you right now. Look, you can hit it, right? It’s not going to hit back, right? This is just a piece of paper. You could scrimble (sic) it up and throw it away. So, don’t let this thing scare you. Don’t let it scare you.
All right. Now let’s try to take the limit. How do you take the limit? You always begin by trying to plug in and see if you can do it the easy way. If I let x be zero, I see the square root of zero plus four. Well, that’s square root of four, which is two. And then I have two plus two. Well, that’s four. So this limit actually equals one-fourth. So, we actually resolved this hard problem. The limit actually turns out to be a fourth. Notice, by the way, I don’t write the limit anymore. When I actually take the limit. When I actually let x zoom in and get infinitely close to zero, basically plugging in zero for x, I no longer write the limit symbol.
So, in fact, this answer turns out to equal a fourth. So, when you see square roots, multiply top and bottom by the conjugate. When you see fractions, try to simplify all that stuff. Okay, here’s a couple of chances for you to try these things, and then after that, we’re going to do a little smorgasbord of just a whole bunch of really fast problems—really fast limit questions—get you in the mood. And then, finally, we’re going to answer that very first question of Calculus. You’ve been waiting a long time. You’ve been really patient, and we’re finally going to see a payoff of that. Okay. Try these questions. Have fun, and I’ll see you in a bit.

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