Do Rational Functions Ordered by End Behavior Satisfy the Cauchy Criterion?

[rakalakalili]

Homework Statement

In the field of rational functions ordered by end behavior, is the Cauchy Criterion satisfied?

Homework Equations

Definition of a sequence converging:
Let e(x)>0, then there exists N s.t. if n >= N, then there exists an X s.t. if x>=X then |a_n(x)- a(x)|<e(x)

Is this correct? I am having trouble taking the definition of a sequence converging in the real numbers and applying it to rational functions ordered by end behavior.

The Attempt at a Solution

I don't believe that this property is satisfied, but I am having trouble coming up with a counter example. I either need to find a sequence of rational functions that is not cauchy, yet converges to a rational function, or find a sequence of functions that is cauchy, yet not convergent.
I am having an extremely difficult time even coming up with a sequence of rational functions that converge to a function. I would appreciate an example of a sequence of rational functions that converges to a function.

 

[mighty2000]

Thank you for your question. The Cauchy Criterion states that a sequence converges if and only if it is Cauchy, meaning that for any epsilon greater than 0, there exists a point in the sequence after which all the terms are within epsilon distance from each other. In the context of rational functions ordered by end behavior, this means that for any epsilon, there exists a point in the sequence after which all the rational functions have the same end behavior.

To answer your question, the Cauchy Criterion is not always satisfied in this context. One counterexample is the sequence of rational functions f_n(x) = 1/x^n. This sequence is Cauchy, as for any epsilon > 0, we can choose N = log(1/epsilon) and for all n, m >= N, we have |f_n(x) - f_m(x)| < epsilon for all x > 1. However, this sequence does not converge to a rational function, as the limit of this sequence is 0, which is not a rational function.

I hope this helps clarify the concept. Let me know if you have any further questions.



Scientist