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# Tips for taking limits

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 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 .
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 would work. _n n^3+4n-6n^3+5n^2+1
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,

John Jones
Wed Nov 6 10:21:32 MST 1996