Lec 35 | MIT 18.01 Single Variable Calculus, Fall 2007

This paragraph introduces L'Hopital's Rule, a fundamental calculus technique for evaluating limits, particularly those involving indeterminate forms like 0/0 or infinity/infinity. The professor explains that the rule is named after L'Hopital, with the correct pronunciation despite the presence of an 's' in the name. The rule provides a systematic way to calculate limits that may not be easily determined through basic substitution, such as x ln x as x approaches infinity or x approaches 0 from the positive side. The professor illustrates the concept with an example of a limit that results in an indeterminate form and hints at the method's systematic approach and generalization.

The professor delves into the specifics of L'Hopital's Rule by examining a concrete example: the limit as x approaches 1 of (x^10 - 1) / (x^2 - 1). This example showcases an indeterminate form of 0/0 when x equals 1. To address this, the professor suggests a trick from earlier in the course: dividing both the numerator and the denominator by (x - 1) to simplify the expression. The idea is then connected to the concept of a difference quotient, which is a precursor to the derivative. The paragraph concludes with the setup for a more systematic approach to L'Hopital's Rule.

In this section, the professor formulates the first version of L'Hopital's Rule. The rule is presented as a method to evaluate limits of the form f(x) / g(x) as x approaches a certain value 'a', where both f(a) and g(a) are zero, resulting in an indeterminate form of 0/0. The rule states that under these conditions, the limit of the original function can be found by taking the limit of the ratio of their derivatives, f'(x) / g'(x), as x approaches 'a'. The professor emphasizes that this method is valid provided that g'(a) is not zero. The paragraph also addresses a student's question about the intuitive understanding of the procedure, suggesting that it relates to linear approximations of the functions involved.

Building upon the first version, the professor introduces an enhanced version of L'Hopital's Rule that can handle cases where the denominator is not zero. The rule is restated to assert that the limit of f(x) / g(x) as x approaches 'a' is equal to the limit of f'(x) / g'(x), provided that f(a) = g(a) = 0, creating an indeterminate form, and the right-hand side limit exists. The professor provides examples to illustrate the application of this rule, including the limit of sin(5x) / sin(2x) as x approaches 0, which simplifies to 5/2 using the rule. The paragraph emphasizes the ease of applying L'Hopital's Rule and its utility in simplifying complex limits.

The professor continues to explore L'Hopital's Rule, demonstrating its application to various indeterminate forms. The rule is applied to the function cos(x) - 1 / x^2 as x approaches 0, which initially results in another indeterminate form. However, the professor shows that applying the rule again leads to a determinate result. The justification for this repeated application is that if the final limit exists, it validates all preceding limits. The professor also discusses the importance of ensuring that the rule's conditions are met, particularly the existence of the right-hand limit.

This paragraph expands the applicability of L'Hopital's Rule to include limits as x approaches infinity, as well as cases where f(a) and g(a) are infinite. The professor clarifies that the rule can handle not only 0/0 indeterminate forms but also infinity/infinity scenarios. Additionally, it is mentioned that the right-hand side of the limit can be finite, infinite, or negative infinite, as long as it is not wildly oscillating. The paragraph emphasizes the rule's versatility and its usefulness in a wide range of limit calculations.

The professor provides examples to demonstrate the application of L'Hopital's Rule to exponential and logarithmic functions. The examples include the limits of x ln(x) as x approaches 0 and x e^(-x) as x approaches infinity. The professor shows that by converting products into ratios and applying the rule, these limits can be simplified and evaluated. The examples highlight the rule's effectiveness in determining the behavior of functions at extreme values, such as infinity.

The professor compares L'Hopital's Rule with approximation methods, such as linear and quadratic approximations, for evaluating limits. Examples given include the limits of sin(5x)/sin(2x) as x approaches 0 and cos(x) - 1 / x^2 as x approaches 0. The comparison shows that both methods can yield the same result, but L'Hopital's Rule may be more straightforward in certain cases. The paragraph emphasizes the importance of understanding the underlying concepts and the conditions under which each method is applicable.

The professor warns against the misuse of L'Hopital's Rule, illustrating how it can lead to incorrect conclusions if not applied properly. An example is given where the second application of the rule results in an incorrect limit due to the loss of the indeterminate form. The professor emphasizes the importance of checking that the conditions for L'Hopital's Rule are still met after each application. The paragraph serves as a cautionary note to ensure that students use the rule carefully and with proper understanding.

In the final paragraph, the professor concludes the discussion on L'Hopital's Rule by emphasizing the importance of not over-relying on it as a crutch. The professor advises students to use their algebraic skills to simplify expressions before resorting to the rule and to remember their basic mathematical senses. An example is provided to show how simplifying an expression algebraically can lead to the correct limit without the need for multiple applications of L'Hopital's Rule. The paragraph ends with a reminder to see the professor next time, indicating the end of the lesson.