If $\lim\limits_{x \to \infty} f'(x) = L$ and $\lim\limits_{n \to \infty} f(n) = A$ exists, prove that $L = 0$. Here is the homework problem I am stuck on:


  Let $f$ be differentiable on $(0,\infty)$.  If $\lim\limits_{x \to \infty} f'(x) = L$ exists in $\mathbb{R}$ and $\lim\limits_{n \to \infty} f(n) = A$ exists in $\mathbb{R}$, prove that $L = 0$.


From the given information, I know that we get to assume:


$f$ is continuous at every point $s \in (0,\infty)$.
We can now apply the MVT.


So far this is what I'm thinking:
First, I think this describes a function which increases towards a horizontal asymptote.
I have created a strictly increasing sequence $\{x_n\} = \{x_1, x_2, \dots, x_n \}$ to serve as the $x$ values in $(0, \infty)$.  This gives me a sequence of intervals basically.

I can apply the MVT on each of these intervals, getting a sequence of $c_n \in (x_{n-1}, x_n)$.  

But I don't see where to go from here.  I am guessing I will need the squeeze theorem later, but there's a gap in between.

Perhaps I'm doing something wrong?

Thanks for your help.
If we can show that $|L|\lt \epsilon$ for every $\epsilon \gt 0$ then we will have that $|L|\le 0\Rightarrow L=0$.

Let $\epsilon\gt 0$ be given. Since $$\lim_{x\to \infty} f^{\prime}(x)=L$$ we can find $M$ such that $x\ge M\Rightarrow |f^{\prime}(x) - L|\lt \frac{\epsilon}{2}$. Since $$\lim_{n\to\infty}f(n)=A$$ we can find $N$ such that $n\ge N\Rightarrow |f(n)-A|\lt \frac{\epsilon}{4}$.

Consider $m\gt max(M,N)$. The mean value theorem tells us we can find $c\in (m,m+2)$ such that $$f^{\prime}(c) = \frac{f(m+2) - f(m)}{2}$$ Then $|f^{\prime}(c)| = |\frac{f(m+2) - f(m)}{2}|\lt |f(m+2)-f(m)|\le |A+\frac{\epsilon}{4} - (A-\frac{\epsilon}{4})|=\frac{|\epsilon|}{2}=\frac{\epsilon}{2}$

Now, $|f^{\prime}(c)-L| = |L-f^{\prime}(c)|\lt\frac{\epsilon}{2}\Rightarrow -\frac{\epsilon}{2}\lt L-f^{\prime}(c)\lt \frac{\epsilon}{2}\Rightarrow f^{\prime}(c)-\frac{\epsilon}{2}\lt L\lt f^{\prime}(c)+\frac{\epsilon}{2}$.

But $-\frac{\epsilon}{2}\lt f^{\prime}(c)\lt \frac{\epsilon}{2}$ so $-\epsilon = \frac{-\epsilon}{2} + \frac{-\epsilon}{2}\lt f^{\prime}(c)+\frac{-\epsilon}{2}\lt L\lt f^{\prime}(c)+\frac{\epsilon}{2}\lt \frac{\epsilon}{2} + \frac{\epsilon}{2}=\epsilon$.

Therefore, $|L|\lt \epsilon$ and must be equal to $0.$
The proof can be done by contradiction. 

Assume that $L\neq 0$, and without loss of generality you can assume that $L&gt;0$ (otherwise use $g(x)=-f(x)$). Since $\lim_{x\to\infty}f(x)=L&gt;0$, for small enough $\epsilon&gt;0$ such that $L-\epsilon&gt;0$, there exist $N&gt;0$ such that for all $x&gt;N$ $f'(x)&gt;L-\epsilon&gt;0$. 

Now we have a differentiable function on an open interval $(N,\infty)$ such that $f'(x)&gt;0$ we can conclude that $f$ is increasing, but this would imply that $\lim_{x\to\infty}f(x)=\infty$ obtaining a contradiction.