INDETERMINATE FORMS AND IMPROPER INTEGRALS CHAPTER 8 INDETERMINATE FORMS AND IMPROPER INTEGRALS

8.1 Indeterminate Forms of Type 0/0 Sometimes when a limit is taken, the answer appears to be 0/0. What does this mean? It may mean several things! To determine what it really does mean, we use a rule developed by l’Hopital in 1696.

Common error: Students eagerly apply l’Hopital’s rule to rational functions that are NOT of the form 0/0! Find the following limit:

Cauchy’s Mean Value Theorem Let f & g be differentiable functions on (a,b) and continuous on [a,b]. If g’(x) does not equal 0 for all x in (1,b), then there is a number c in (a,b) such that:

8.2 Other Indeterminate Forms L’Hopital’s rule also holds true for the following indeterminate forms:

Other indeterminate forms Take the logarithm and then apply l’Hopital’s in the following cases:

Example

8.3 Improper Integrals: Infinite Limits of Integration Improper integrals: If the limits of integration, a and b, are one, the other, or both equal to infinity.

Definition If the limits on the right exist and have finite values, then the corresponding improper integrals converge and have those values. Otherwise, the integrals diverge.

Definition If the integral of a function from negative infinity to 0 converges and the integral of the same function from 0 to infinity converses, that the function integrated from negative to positive infinity also converges to the sum of the 2 integrals:

8.4 Improper Integrals: Infinite Integrands Let f be continuous on the half-open interval [a,b) and suppose that Provided that this limit exists and is finite, in which case we say it converges. Otherwise, it diverges.

Example

Integrands that are infinite at an interior point Let f be continuous on [a,b] except at a number c, where a<c<b, and suppose that provided both integrals on the right converge. Otherwise, the integral diverges.