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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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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.

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Rational function and polynomials

The reason why polynomials are so important is that there is a theorem from Analysis that says that any continuous function defined on an interval of the real line can be approximated arbitrarily closely by a polynomial. So polynomials are useful to ¿model¿ any kind of function on a closed interval. However, polynomials ¿get wild¿ at infinity, so they don¿t work well to try to extrapolate an arbitrary function past the closed interval in which it is being approximated by the polynomial.

A rational function is a function which is a ratio of two polynomials, one polynomial in the numerator and another one in the denominator. Rational functions are also used to model an arbitrary function, and for many purposes they have better behavior. If the rational function is a ratio of two polynomials of the form p(x)/q(x), and the order of the two polynomials is np and nq, try to give a qualitative description of the behavior of this rational function. What happens to the rational function in the cases np > nq, np = nq, and np < nq as x goes to plus or minus infinity (compare with the case of a polynomial)? If an arbitrary function f(x) goes to zero at plus and minus infinity, what kind of rational function would be best to model this function?

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