Periodic Functions f(x)=r(x)/s(x)

Question

I have two questions in one.

(1) Let $f(x)=\frac{s(x)}{r(x)}$. If $s(x)$ and $r(x)$ are polynomials of same degree, prove that if $f(x)$ is periodic, it must be constant for all $-\infty<x<\infty$

I'm struggling with the proof. I know that given $f(x)$ is periodic, $\frac{s(x)}{r(x)}=\frac{s(x+p)}{r(x+p)}$. Clearly the antiderivative of $f(x)$ I can take to be constant on $x$ to $x+p$, but that doesn't help me with understanding the importance of the same degree polynomials.

(2) Let $f(x)=\frac{s(x)}{r(x)}$. Now if $s(x)$ and $r(x)$ have non-equal degree, show $f(x)$ cannot be periodic.

Please help me understand the importance of our polynomial degrees. Thanks

Answer 1

Hint

If $r(x)$ and $s(x)$ are polynomials of the same degree then, what can you say about $\lim_{x\to \infty} \frac{r(x)}{s(x)}?$ Isn't it a real number? Now, can a non-constant periodic function have limit as $x\to \infty?$

If $r(x)$ and $s(x)$ are polynomials of different degree then $\lim_{x\to \infty} \frac{r(x)}{s(x)}=\pm \infty$ if the degree of $r$ is bigger than the degree of $s.$ Can a periodic function satisfy that? If the degree of $r$ is smaller than the degree of $s$ $\lim_{x\to \infty} \frac{r(x)}{s(x)}=0.$ Can a non-constant periodic function satisfy that? Can it be constant?

Answer 2

Hint. If $p$ is a period of $f$ then, as you wrote, you have $$ \frac{s(x)}{r(x)}=\frac{s(x+p)}{r(x+p)}, \quad x \in \mathbb{R}, $$ giving, for any integer $n$, $$ \frac{s(x)}{r(x)}=\frac{s(x+n\times p)}{r(x+n\times p)}, \quad x \in \mathbb{R}, $$ and, as $n \to +\infty$, we obtain $$ \frac{s(x)}{r(x)}=\frac{s(x+n\times p)}{r(x+n\times p)} \longrightarrow \frac{a_m}{b_m},\qquad x \in \mathbb{R}, $$ or $$ \frac{s(x)}{r(x)}=\frac{a_m}{b_m},\qquad x \in \mathbb{R}, $$ where $$ \begin{align} s(x)&=a_m x^m+a_{m-1}x^{m-1}+...+a_0\\ r(x)&=b_m x^m+b_{m-1}x^{m-1}+...+b_0. \end{align} $$