Indeterminate Forms and L

Indeterminate Forms and L - Indeterminate Forms and...

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Indeterminate Forms and L’Hospital’s Rule Back in the chapter on Limits we saw methods for dealing with the following limits. In the first limit if we plugged in we would get 0/0 and in the second limit if we “plugged” in infinity we would get ( recall that as x goes to infinity a polynomial will behave in the same fashion that it’s largest power behaves). Both of these are called indeterminate forms . In both of these cases there are competing interests or rules and it’s not clear which will win out. In the case of 0/0 we typically think of a fraction that has a numerator of zero as being zero. However, we also tend to think of fractions in which the denominator is going to zero as infinity or might not exist at all. Likewise, we tend to think of a fraction in which the numerator and denominator are the same as one. So, which will win out? Or will neither win out and they all “cancel out” and the limit will reach some other value? In the case of we have a similar set of problems. If the numerator of a fraction is going to infinity we tend to think of the whole fraction going to infinity. Also if the denominator is going to infinity we tend to think of the fraction as going to zero. We also have the case of a fraction in which the numerator and denominator are the same (ignoring the minus sign) and so we might get -1. Again, it’s not clear which of these will win out, if any of them will win out. With the second limit there is the further problem that infinity isn’t really a number and so we really shouldn’t even treat it like a number. Much of the time it simply won’t behave as we would expect it to if it was a number. To look a little more into this check out the Types of Infinity section in the Extras chapter at the end of this document. This is the problem with indeterminate forms. It’s just not clear what is happening in the limit. There are other types of indeterminate forms as well. Some other types are,
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These all have competing interests or rules that tell us what should happen and it’s just not clear which, if any, of the interests or rules will win out. The topic of this section is how to deal with these kinds of limits.
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