Eli5 graphing exponential growth/decay with rational functions

Question

I understand each individual aspect of rational functions, but as soon as it comes to the growth and decay I have no clue what I’m doing. Something isn’t clicking for me; what steps should I be following?

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Answer 1

I think you may need to explain in more detail what you are trying to do. There are many different ways in which you could approximate exponential functions with rational functions, but you will never capture the actual long-term exponential growth/decay. If f(x) is a rational function, then for large x it will behave like a x^n for some real a and integer n, which is slower than exponential.