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Functions and infinity

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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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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)?

CASE1

When order of numerator is greater than denominator i.e. np > nq, the numerator grows at a rate that is much higher than the denominator. Hence, the rational function will tend to + infinity or - infinity depending on the signs.

For example,

let p(x) = 3x^2 + 2 where ^ means power.
let q(x) = x + 1

np =2 amd nq = 1 here.

As x tends to + infinity 3x^2 + 2 grows much faster in magnitude than x + 1. Therefore, the ratio tends to infinity.

In mathematical terms, we can see that 3x^2 + 2/(x + 1) = [3x + 2/x]/(1 + 1/x)

As x tends to infinity, 1/x and 2/x tends to 0. Hence p(x)/q(x) lim x -> infinity = 3x which tends to infinity as x becomes
large on postive side.

If x tends to - infinity, then the numerator tends to + infinity since square is always positive. On the otherhand, x + 1 tends to - ...

Solution Summary

This provides explanations of what happens to certain functions as they approach infinity.

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