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Calculus: The Limit Laws, Part I


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About this Lesson

  • Type: Video Tutorial
  • Length: 2:31
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 26 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: The Concept of the Limit (8 lessons, $12.87)

In this lesson, you will learn about the limits of functions. Since limits are real numbers, you can apply many of the properties of real numbers to limits. Professor Burger walks you through many of these, including: limits of function combinations (limits of the sum of two functions, limits of the product of two functions, limits of the quotient of two functions) and limits of functions times a constant.

For Part II of Professor Burger's limit law lessons, check out

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 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

2174 lessons

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 or visit Thinkwell's Video Lesson Store at

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


Recent Reviews

Good refresher
~ jfilip

This lesson is short, but it covers limit laws which is what I was looking for. I don't know how useful it would be alone but as a part of the series it fits perfectly. Limits came rushing back from my high school days.

Good refresher
~ jfilip

This lesson is short, but it covers limit laws which is what I was looking for. I don't know how useful it would be alone but as a part of the series it fits perfectly. Limits came rushing back from my high school days.

The Concept of the Limit
The Limit Laws Page [1 of 1]
Okay, so we’re taking these limits and you can see how we can actually approach a function and see what the function is heading towards as the x’s get closer and closer to whatever it is that we want the x’s to get closer and closer to.
Now, in fact, once you have the limit of one function, you can put them together in all sorts of interesting ways, and the results, not surprising, can be combined. So let me just show you. Suppose we have the limit as x ? a of f(x), and suppose that that equals L. And suppose I have a different function, limit, as x ? a of some other function, let’s say g(x), and that equals M. So both limits exist. They both are heading toward a target.
Now, the question is, well, what can be said about combinations of these functions? Well, these are just limit laws that I think make a lot of since. For example, suppose you want to take the limit as x ? a of the sum of these two functions? What would you guess that limit would be? That’s right, I heard you, it’s going to be just the sum of the limits. And what if I look at the difference? If I look at the difference of the two functions, then you just look at the difference of the answers. So that makes a lot of sense.
What if you take their product? So let’s try to be really, really fancy. x ? a of f(x) and you multiply by g(x). Well then, in fact, what you get would be just the product of the limits. In fact, you could almost see the word “limit” right there. If I put a little “i” in there, “limit”! That’s perfect. Okay, great!
All right, now what about taking the quotient? Well, taking the quotient, we’ve got to be a little bit more careful, but not too much more careful. This will equal , but we’ve got to make sure, of course, that we’re not dividing by 0. So I want to make sure that, in fact, this is going to be where M ? 0, because if M = 0, then, of course, this limit wouldn’t make a lot of sense, because I’d have 0 downstairs and who know what that would be. So I want to actually make sure that both limits exist in this case and the denominator ? 0. Otherwise, you can just plow away and do limits as you would normally think.
Oh, and by the way, here’s another one. In fact, make up your own. There’s a great thing; make up your own little limit law. Here’s one I’ll make up right now live. Suppose I give you just some number, I’ll call it c for a constant, and I take the limit of this. Well, what do you think it is? Well, since c is fixed and unchanging and I know that this thing is heading toward L, then I would guess this goes to c × L, and that’s the right answer. So you can make up your own limit laws and have a lot of fun with them.
Anyway, here’s some basic facts that I think are pretty intuitive, but now we’ve seen them. I’ll see you at the next lecture.

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