Question: > 
> Suppose $f:\mathbb{R} \to \mathbb{R}$ is a differentiable function such that $\lim\limits\_{x\to\infty}f'(x)=1$,then is it true necessarily true that $f(x)$ unbounded?
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I think that it will always intersect $y=c$ for every $c\in \mathbb{R}$ and thus cannot be bounded!

Answer: 
If the derivative tends to 1 then in particular it will eventually become bigger than 1/2. Therefore the function is dominated from below by $\frac{1}{2}x-c$ for a suitable c, and hence tends to infinity.

More precisely, by definition of limit for every $\epsilon>$ there is an $N$ such that if $x>N$ then $f'(x)>1-\epsilon$. Now choose $\epsilon=1/2$. Then for every $x>N$ we have $f'(x)>1/2$. Now proceed as above.
Choose some $c\in\left(0,1\right)$. 

$\lim\_{x\rightarrow\infty}f'\left(x\right)=1$
implies the existence of some $x\_{0}$ with $x\geq x\_{0}\Rightarrow f'\left(x\right)\geq c$.

Then $f\left(x\right)\geq cx+f\left(x\_{0}\right)-cx\_{0}$ for $x\geq x\_{0}$
showing that $f$ is unbounded.