Two basic ways to compare the size of two quantities is to take their difference or ratio. We'll focus on the ratio since relative size is important.

Can we find the ratio of two functions when the values of both of those functions are tending toward zero? What would this correspond to geometrically?

Can we find the ratio of two functions when the values of both of those functions are blowing up to infinity? What would this correspond to graphically?

Rational functions are quotients of polynomials, so naturally they compare the relative size of the polynomial in the numerator to the polynomial in the denominator using a ratio. We can also take ratios of other types of functions.