In this question it is proved in the answers that If $f :[0 , \infty ) \to \mathbb{R}$ and $f $ is bounded on every $(a,b)$ such that $a<b <\infty$, prove that $\lim\limits_{x \to \infty }(f(x+1) - f(x) )=l$ implies $\lim\limits_{x \to \infty }\frac{f(x)}{x}=l$.
This condition looks suspiciously similar to Stolz-Cesaro theorem see this for a proof of using Stolz-Cesaro theorem .
My question is : If If $f,g :[0 , \infty ) \to \mathbb{R}$ $\lim\limits_{x\to\infty}\frac{f(x+1)-f(x)}{g(x+1)-g(x)}=l$ what are sufficient conditions for $f,g$ to make $\lim\limits_{x\to\infty}\frac{f(x)}{g(x)}=l$?
Of course when $f$ and $g $ are both differentiable functions on $\mathbb{R}$ and if $\lim\limits_{x \to \infty } g(x)= \infty$ this condition sufficient to make $\lim\limits_{x\to\infty}\frac{f(x)}{g(x)}=l$, but this hypothesis is too strong is there a weaker hypothesis. and I want to generalise this result.
I conjectured this more general hypothesis : If $f$ is bounded on every $(a,b)$ such that $a<b <\infty$, $g$ is a continuous increasing function on $\mathbb{R}$ then $\lim\limits_{x\to\infty}\frac{f(x)}{g(x)}=l$.
I couldn't prove this result but I think it is true.
My question is this conjecture true ? If it was true is there a weaker hypothesis ? If it wasn't true, Is there a weaker hypothesis than $f$ and $g $ are both differentiable functions on $\mathbb{R}$ and ]$\lim\limits_{x \to \infty } g(x)= \infty$?
