What about rational functions creates undefined and asymptotic behavior?

Answer 1

A rational function is the ratio of two polynomial functions. The function that is the denominator will have roots (or zeros) in the complex field and may have real roots. If it has real roots, then evaluating the rational function at such points will require division by zero. This is not defined.

Since polynomials are continuous functions, their value will be close to zero near their roots. So, near a zero, the rational function will entail division by a very small quantity and this will result in the asymptotic behaviour.