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
happens

- when the numerator and denominator have the same
degree (
*s*=*t*), the limit is 1; - when
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