Presentation on theme: "INDETERMINATE FORMS AND IMPROPER INTEGRALS"— Presentation transcript:
1 INDETERMINATE FORMS AND IMPROPER INTEGRALS CHAPTER 8INDETERMINATE FORMS AND IMPROPER INTEGRALS
2 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.
3 Common error: Students eagerly apply l’Hopital’s rule to rational functions that are NOT of the form 0/0!Find the following limit:
4 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:
5 8.2 Other Indeterminate Forms L’Hopital’s rule also holds true for the following indeterminate forms:
6 Other indeterminate forms Take the logarithm and then apply l’Hopital’s in the following cases:
8 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.
9 DefinitionIf the limits on the right exist and have finite values, then the corresponding improper integrals converge and have those values. Otherwise, the integrals diverge.
10 DefinitionIf 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:
11 8.4 Improper Integrals: Infinite Integrands Let f be continuous on the half-open interval [a,b) and suppose thatProvided that this limit exists and is finite, in which case we say it converges. Otherwise, it diverges.
13 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 thatprovided both integrals on the right converge. Otherwise, the integral diverges.