Rational Functions: Degree of Denominator vs Nominator

[Niles]

Hi all.

I have always wondered: If we e.g. look at functions given by

$$f(x) = \frac{\cos x}{x^2}, \quad g(x) = \frac{\sin x}{x^2}, \quad h(x) = \frac{\exp x}{x^2},$$
then does the degree of the denominator exceed the degree of the nominator by 1 or by 2?

 

[Count Iblis]

These are not rational functions. The numerator could be assigned a degree of infinity as they are analytic functions that are not polynomials.

 

[Niles]

Thank you.