What is the logic behind the method for finding horizontal asymptote of a rational function?

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I know the methods perfectly well but I don't understand why they are the way they are. For example, finding the vertical asymptote makes perfect sense as setting the denominator to 0 just finds you the values of x that undefines the function. However, I don't see any logic behind the methods to find the horizontal asymptotes and it doesn't make any intuitive sense to me.