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**INDETERMINATE FORMS AND IMPROPER INTEGRALS**

CHAPTER 8 INDETERMINATE FORMS AND IMPROPER INTEGRALS

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**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.

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

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**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:

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**8.2 Other Indeterminate Forms**

L’Hopital’s rule also holds true for the following indeterminate forms:

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**Other indeterminate forms**

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

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Example

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**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.

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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.

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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:

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**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.

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Example

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**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.

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