I was curious about this exercise, because I thought it could be a valuable tool to use the theorem dell'asintoto ... I do not think that is the way, does anyone have any idea? !
Let $ h $ is a function defined on $(a, + \infty)$ and limited all intervals $(a, b)$
$a <$ b. Prove that if $$ \lim_ {x \to + \infty} [h (x +1)-h (x) ] = k, $$
then $$\lim_ {x \to + \infty} \frac {h (x)}{x} =k$$