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The limits which arise from the ratio test often contain
rational functions of n. In other words, one has to
compute limits of the sort
Since we are interested in the behavior of the fraction when
n is large, we suspect that the lower power terms do not
affect the outcome and that the result is the same as for
. This is exactly what
These rules both make sense in terms of the growth rates for
different powers of n, and can be justified with algebra.
We give one example to illustrate how an algebraic proof
- when the numerator and denominator have the same
degree (s=t), the limit is 1;
the degree of the numerator, s, is less than the degree of
the denominator, t, the limit is 0;
- when the degree of the numerator, s, is greater than the
degree of the denominator, t, the limit is .
The main idea is to multiply both top and bottom by the same
quantity, in this case . Then, the limit of
each part of the expression is easy to compute individually.
The rules listed above can all be justified in general with
this technique. In applying the ratio test, feel free to
use the rules listed above without further explanation.
Finally, we have described how one may determine limits of
rational functions of n when the numerator and
denominators have leading coefficients of 1. While other
cases are unlikely to arise from the ratio test, they are
not any more difficult to compute. For example,
Wed Nov 6 10:21:32 MST 1996