Determination of limit of a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) = f(x + 2)$ for every real number $x$

Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a nonconstant function satisfying $f(x + 2) = f(x)$ for every real $x$. Then $\lim_{x \to \infty} f(x)$

(A) does not exist.

(B) exists and equals $+\infty$ or $−\infty$.

(C) exists and is finite.

(D) may or may not exist depending on $f$.

Answer 1

The answer is (A). To prove this, let $x,y$ be reals such that $f(x)\neq f(y)$. Then, set $x_n=x+2n$ and $y_n=y+2n$. What happens when we take the limit of $f$ applied to each subsequence?