If $f:R\to R$ is bounded on bounded intervals, and $\lim f(x+1) - f(x) = L$ for $x\to +\infty$ then $\lim f(x)/x = L$ when $x\to +\infty$
If $\lim_{x\to+\infty}[f(x+1)-f(x)]= \ell,$ then $\lim\limits_{x\to+\infty}\frac{f(x)}x=\ell$.