<s>[INST] If $ \lim_{x\to \infty} (f(x+1)-f(x))=1$, then $ \lim_{x\to \infty} \frac{f(x)}{x}=1$? Let $f:\mathbb{R} \to \mathbb{R}$ be a fuction such that $\displaystyle \lim_{x\to \infty} (f(x+1)-f(x))=1$. Is it true then that $\displaystyle \lim_{x\to \infty} \frac{f(x)}{x}=1$?
I think it is and here is how I went about it. Let $\varepsilon>0$. Then there is some $\delta_\varepsilon>0$ such that $$1-\varepsilon < f(x)-f(x-1)<1+\varepsilon, \forall x>\delta_\varepsilon.$$
Thus, we may write the following inequalities for an $x> \delta_\epsilon$:
$$1-\varepsilon < f(x)-f(x-1)<1+\varepsilon\\
1-\varepsilon < f(x-1)-f(x-2)<1+\varepsilon\\    
\vdots\\
1-\varepsilon < f(\delta_\varepsilon+1)-f(\delta_\varepsilon)<1+\varepsilon$$
and after we sum these up we get that $$(x-\delta_\epsilon)(1-\epsilon)<f(x)-f(\delta_\varepsilon)<(x-\delta_\varepsilon)(1+\varepsilon), \forall x>\delta_\varepsilon.$$
This implies that $$\frac{f(\delta_\varepsilon)+(x-\delta_\varepsilon)(1-\varepsilon)}{x}<\frac{f(x)}{x}<\frac{f(\delta_\varepsilon)+(x-\delta_\varepsilon)(1+\varepsilon)}{x}, \forall x>\delta_\varepsilon.$$
If we take $\displaystyle\limsup_{x\to\infty}$, we get that $1-\varepsilon < \displaystyle\limsup_{x\to\infty} \frac{f(x)}{x}< 1+\varepsilon$, $\forall \varepsilon > 0$, so $\displaystyle\limsup_{x\to\infty} \frac{f(x)}{x}=1$. In the same way we get that $\displaystyle\liminf_{x\to\infty} \frac{f(x)}{x}=1$, so $\displaystyle\lim_{x\to\infty} \frac{f(x)}{x}=1$ as desired.
Is this proof correct? I am a bit unsure that I am taking that $\limsup$ correctly, even though I can't see why it could be wrong.
 [/INST] Let
$
f(x)=
-\sin \left( \lfloor x\rfloor\cdot \dfrac{\pi}{2} \right) 
\cdot
 \cos\left( \lfloor x\rfloor\cdot \dfrac{\pi}{2} \right). 
$
Then
\begin{align}
f(x+1)-f(x)
=&
\sin \left( \lfloor x+1\rfloor\cdot \dfrac{\pi}{2} \right) 
\cdot
 \cos\left( \lfloor x+1\rfloor\cdot \dfrac{\pi}{2} \right)
-
\sin \left( \lfloor x\rfloor\cdot \dfrac{\pi}{2} \right) 
\cdot
 \cos\left( \lfloor x\rfloor\cdot \dfrac{\pi}{2} \right)
\\
=&+\sin\left((\lfloor x+1\rfloor - \lfloor x\rfloor)\cdot \dfrac{\pi}{2}\right)
\\
=& +\sin\left( 1\cdot \frac{\pi}{2} \right)
\\
=& 1
\end{align}
and  by $-1\leq f(x)=-
\sin \left( \lfloor x\rfloor\cdot \dfrac{\pi}{2} \right) 
\cdot
 \cos\left( \lfloor x\rfloor\cdot \dfrac{\pi}{2} \right)\leq +1$ we have
$$
\lim_{x\to 0}\frac{f(x)}{x}=0.
$$
</s>