I don’t know how to formally prove that if $$\lim_{x \to \infty} \left(f(x+1)-f(x)\right) = L,$$ then $$\lim _{x \to \infty} \frac{f(x)}{x} = L$$
where L is a constant and the function is limited whit in a limited domain. I tryed to expand the definition of infinit limit but have no idea where to go on from there or even if that is the best aproach.