Does a function $f$ converges at $+\infty$ if all $f(n+t)_{n\in\mathbb{N}}$ converge to the same limit?
Let $f$ be a continuous function satisfying $\lim \limits_{n \to \infty}f(x+n) = \infty$ for all $x$. Does $f$ satisfy $f(x) \to \infty$?