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Function of polynomial ratios

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Now consider a rational function, which is the ratio of two polynomials. These two polynomials will each have a set of zeros, and note that at a zero of the denominator we are actually dividing by zero. The zeros of the denominator are called poles, and they are point where the rational function becomes infinite (unless there is also a zero of the numerator at that point, which is called a zero of the function).
Describe the behavior of a rational function near one of its poles on the real axis. How does the function vary as the pole is approached from each side, and what happens at the pole? If there is also a zero at this point, then what happens at the pole? We know that zero divided by zero is indeterminate, but if there is a zero and a pole at the same point, what is a sensible definition of the function at this point?

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Solution Preview

A rational function of variable x can be written as p(x)/q(x) where p and q are both polynomial functions of x.

Describe the behavior of a rational function near one of its poles on the real axis?
As x approaches a zero of q(x), the actual value of q(x) becomes very small. Since q(x) is a continuous function, we can approach as close to the zero as we want and hence q(x) can become arbitrarily small in magnitude. Hence, p(x)/q(x) will approach + infinity or -infinity as this happens depending on the sign of p(x) on the side from which we ...

Solution Summary

This provides an example of giving a definition for a function that is the ratio of polynomials.