Asymptotes are one way - not the only way, but one of several - to analyze the general behavior of a function.
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Nope not all the rational functions have a horizontal asymptote
That's the definition of a "rational function". You simply divide a polynomial by another polynomial. The result is called a "rational function".
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A rational number can always be written as (one whole number) divided by (another whole number). That's not only a characteristic of a rational number, it's the definition of one.