**Asymptotic behavior of rational functions** is a relation between the time-dependence and the behavior of an abstract machine. A machine that behaves asymptotically is one in which the output of a set of inputs takes the same time to converge as the input as it does to get from its input. In asymptotic behavior of rational functions, the same inputs take the same time to reach their output as they did to get from their input. This is sometimes called as the converging process of asymptotic behavior of rational functions. The definition of asymptotic behavior of rational functions was first developed by Fischer (1947).

The study of **asymptotic behavior of rational functions** goes on using a number of techniques. A prime example of asymptotic behavior of rational functions is used in Bell (19ior), who introduced the measure units to be used in Bell’s technique. According to Bell, if we replace the notation of units with the notation of measures we get the asymptotic behavior of rational functions as the function satisfying the inequality stated in the set-valued function is called the local function. According to Bell, if we replace the notation of measures with the notation of measures which are not set, but which are assumed to be real, then the inequality Bell calls the local function may be called the Bell’s inequality.

Another widely used approach to analyze the asymptotic behavior of rational functions is to use random variables. These variables may be real or artificial. To study the asymptotic behavior of rational functions in random variables, we need to make sure that they do not have prior conditions that affect their behavior. This ensures that the asymptotic behavior of a rational function is completely given by its random variables. Theorems like the Bell’s Theorem and also the techniques used in set theory such as the lattices and cohomology can be used to study the asymptotic behavior of rational functions in random variables.

A few recent developments in computing theory, especially as regards the study of **asymptotic behavior of rational** numbers, has resulted in the formulation of a new theory of asymptotic behavior of rational numbers. This new approach conforms closely to Bell’s Theorems. The main difference is that under this new approach, the notion of the local minima of the sequences of natural numbers itself is neglected. On the other hand, asymptotic behavior of rational numbers in closed discrete systems is studied using Bell’s Theorems and other works on the foundations of mathematics.

Some recent works on asymptotic behavior of rational numbers have been motivated by questions regarding the trustworthiness of computing theory as a whole. One of the problems is related to the axiom of large numbers. According to this axiom, any number smaller than a million may be regarded as a countably infinite number. Another issue concerns the treatment of real numbers as compared with the treatment of digits.

Several recent works deal extensively with **asymptotic behavior of rational** numbers. In one paper, Bell, Kolb, and Zaltman point out that the treatment of digits is not as straightforward as it seems. digits can be transformed into other units as well as back into digits at different times. The combined result of all these arguments indicates that the axiom of large numbers does not correctly describe the real world as we know it.

## Behavior of Rational Functions

Behavior of irrational functions is governed by a dominant logical function. The term “rational” is used here to refer to the set of all subsets of sets (set theory) and their members, namely numbers, elements of order and composition, and sets of objects. According to set theory, there are no properties of the objects which can be described by means of definite descriptions; rather the properties of the objects are wholly qualitative, meaning that their description involves descriptions of their qualitative attributes. Accordingly, as noted, rational behavior of rational functions generally is dominated by a single dominant logical function. This single logical function is called the “principal” function. In this way, according to set theory, rational behavior is essentially just the sum of the values of the principal functions on an unvaried basis.

In the mathematical parlance, the term “behavior of rational functions” denotes the behavior of rational functions under various assumption sets, which are given as simple x-ary models of sets of real numbers (hereafter referred to as real numbers multiplied by prime number). Under such assumption, a polynomial function f(x) is called a rational function r(x), if for every x such that the function x is a monotonic function in the sense of being a conjugate of the corresponding irrational function h(x) such that there is an x such that the rational function f(x) is the xth h(x) -th power of the natural logarithm.

For example, the rational function r(x) of the complex number y(y’=x) can be written as the set of real numbers (whose sum is greater than zero) such that the formula x(y’=x) is valid if y(y’=x) is a natural number. It follows that the value of x(y’=x) of the function f(x) in question is just the value of y(x) when expressed as a real number. If y’ is a real number, then the behavior of rational functions of the function f(x) is called a rational function reversal, where the function f(x) is called from its reversal by the formula y(y’=x) for some real number y such that the value of x’ is greater than zero.

The rational behavior of rational functions in such cases as above can be described as a set of end behaviors of rational functions such that their sum is greater than zero and such that their sum is converging to some definite set of values, which satisfy certain axioms. A particular choice of such a function may satisfy one axiom, but not another, or conversely, it may satisfy two axioms and not the other. We say then that the set of end behavior of rational functions can be called a polynomial collection, and thus we call it the “poodle’s paradox”.