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Calculus: Indeterminate Forms 
Calculus: Introduction to L'Hopital's Rule 
Calculus: Basic Uses of L'Hopital's Rule 
Calculus: Exotic Examples of Indeterminate Forms 
Calculus: L'Hopital's Rule, Indeterminate Products 
Calculus:L'Hopital's RuleIndeterminate Difference 
Calculus: L'Hopital's Rule, 1 to an Infinite Power 
Calculus: Example of One to the Infinite Power
About this Series
 Lessons: 8
 Total Time: 1h 22m
 Use: Watch Online & Download
 Access Period: Unlimited
 Created At: 07/29/2009
 Last Updated At: 06/01/2011
In this 8lesson series, we'll look at indeterminate quotients and other indeterminate forms.
When taking limits, sometimes you will encounter expressions whose meanings can be interpreted in different ways. These limits are called indeterminate forms. A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression. Indeterminate forms are also called indeterminant forms. In this fourlesson series, we'll look at basic indeterminate forms and how L'Hopital's rule helps us to evaluate them. The two types of indeterminate forms we'll focus on are 0/0 and infinity/infinity. Some indeterminate forms can be solved by using algebra tricks such as canceling or dividing by the highest power of x. Some expressions, however are camouflaged indeterminate forms that needs to be manipulated using mathematical identities or properties in order to establish that they are indeterminate.
L'Hopital's rule enables you to evaluate indeterminate forms quickly by using derivatives. When evaluating a limit, it is always a good idea to plug in the value first. If the result yields an indeterminate form, then use L'Hopital's rule. As long as the limit is still an indeterminate form you can reuse L'Hopital's rule. L'Hopital's rule might not produce the right answer if you use it on a limit that does not produce an indeterminate form.
As long as the limit still produces an indeterminate form you can reuse L'Hopital's rule. When applying L'Hopital's rule to a quotient containing one or more products or compositions of functions, it is necessary to use the product or chain rules. L'Hopital's rule, however, might not give you the right answer if you use it on a limit that does not produce an indeterminate form.
Sometimes, we can use mathematical identities or properties to manipulate or restate an expression such that we can verify that it is an indeterminate form. A limit of a function is called an indeterminate form when it produces a mathematically meaningless expression. Indeterminate forms are also called indeterminant forms.
In these lessons, we'll look at indeterminate products (e.g. 0 times infinity) and indeterminate differences. We'll also use properties of logarithms to restate things like 1^infinity in order to find an indeterminate form that we can the apply L'Hopital's rule to.
Taught by Professor Edward Burger, this series comes from a comprehensive Calculus course. 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'Hopital'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.
About this Author
 Thinkwell
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Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
Lessons Included
 Calculus: Indeterminate Forms
 Calculus: Introduction to L'Hopital's Rule
 Calculus: Basic Uses of L'Hopital's Rule
 Calculus: Exotic Examples of Indeterminate Forms
 Calculus: L'Hopital's Rule, Indeterminate Products
 Calculus:L'Hopital's RuleIndeterminate Difference
 Calculus: L'Hopital's Rule, 1 to an Infinite Power
 Calculus: Example of One to the Infinite Power
Below are the descriptions for each of the lessons included in the series:

Calculus: Indeterminate Forms
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. 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'Hopital'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 UTAustin 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, padic 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.

Calculus: Introduction to L'Hopital's Rule
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. 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'Hopital'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 UTAustin 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, padic 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.

Calculus: Basic Uses of L'Hopital's Rule
In this lesson, we will look at applications of L'Hopital's rule, when to use it, what to watch out for, etc. When assessing limits, start by plugging in using substitution to evaluate them. If this gives you an indeterminate form, apply L'Hôpital's rule. The L'Hôpital rule is applicable to f(x) and g(x) if they are differentiable functions across an interval containing c, except possibly at c. If the limit as x approaches c of f(x)/g(x) produces the indeterminant form of 0/0 or (+infinity)/(+infinity), then the limit as x approaches c or f(x)/g(x) is equal to the limit as x approaches c of [the derivative of f(x)] / [the derivative of g(x)] provided the limit on the right exists or is infinite. L'Hôpital's rule can also be applied to onesided limits and, as long as the limit is indeterminate, you can take the limit again if L'Hôpital's rule doesn't give you an answer the first time around (and you are again left with an indeterminate form as the limit). Note that the rule may mislead you if you misapply it to a limit that doesn't produce an indeterminate form.
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 UTAustin 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, padic 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.

Calculus: Exotic Examples of Indeterminate Forms
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. 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'Hopital'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 UTAustin 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, padic 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.

Calculus: L'Hopital's Rule, Indeterminate Products
Some indeterminate forms, including indeterminate products, must be manipulated before L'Hôpital's Rule can be applied. In order to do this, try expressing one of the factors as a fraction (so you can recognize things like (0 * infinity) or (theta * cotangent of theta) as an indeterminate form given that they are actually indeterminate products rather than the easily recognizable 0/0 or infinity/infinity). Once recognized as an indeterminate form, you will learn how to manipulate these camouflaged indeterminate forms in such a way that they can have L'Hôpital's rule applied to them. In this lesson, you will learn how to recognize these limits with indeterminate forms as well as how to handle limits with negative exponents and limits with trigonometric expressions.
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 UTAustin 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, padic 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.

Calculus:L'Hopital's RuleIndeterminate Difference
This lesson will show you how to recognize indeterminate forms that are camouflaged as indeterminate differences rather than showing up as 0/0 or infinity/infinity. For instance, when you calculate the limit to be equal to infinity  infinity (e.g. 1/x  1/(e^x1)), this is an indeterminate form, so you must manipulate it to make it into a form such that you can apply L'Hôpital's rule to it. This lesson also covers additional camouflaged indeterminate differences that show up with fractions and radicals. To apply L'Hôpital's rule to an indeterminate difference in many of these instances, you will first have to find a common denominator or look for a sneaky way to factor. Professor Burger will walk you through examples of all of these instances.
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 UTAustin 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, padic 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.

Calculus: L'Hopital's Rule, 1 to an Infinite Power
The limit of a function is called an indeterminate form when it produces a mathematically meaningless result. Some advanced indeterminate forms have to be manipulated before L'Hôpital's rule can be applied to them. In this lesson, you will learn several ways in which logarithmic functions in indeterminate forms can be camouflaged. In order to use L'Hôpital's rule on a camouflaged indeterminate form (e.g. 1^infinity or infinity * 0), you will need to apply logarithmic properties such that you can rewrite the exponential function as a logarithm and convert it into a standard indeterminate form. L'Hopital's rule is applicable to f(x) and g(x) if they are differentiable functions across an interval containing c, except possibly at c. If the limit as x approaches c of f(x)/g(x) produces the indeterminant form of 0/0 or (+infinity)/(+infinity), then the limit as x approaches c or f(x)/g(x) is equal to the limit as x approaches c of [the derivative of f(x)] / [the derivative of g(x)] provided the limit on the right exists or is infinite. L'Hôpital's rule can also be applied to onesided limits and, as long as the limit is indeterminate, you can take the limit again if L'Hôpital's rule doesn't give you an answer the first time around (and you are again left with an indeterminate form as the limit)
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 UTAustin 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, padic 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.

Calculus: Example of One to the Infinite Power
Taught by Professor Edward Burger, this lesson comes from a comprehensive Calculus course. 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'Hopital'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 UTAustin 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, padic 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.
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Very helpful for me as I plunge forward in Calc 2. For some reason I was making a minor mistake differentiating the ln(f(x)) portion of this problem. Good to always make sure you are placing items correctly in the numerator or denominator. I was working on a similar problem that was the same except it was 3/x instead of 1/x. Of course the answer is still e! Thanks!