$f:\mathbb R \to \mathbb R$ be continuously differentiable function such that $f(x),f'(x)>0$ for all real $x$ , then $\lim _{x \to -\infty}f'(x)=0$? Let $f:\mathbb R \to \mathbb R$ be a continuously differentiable function such  that $f(x)>0 , f'(x)>0 , \forall x \in \mathbb R$ , 
then is it true that $\lim _{x \to -\infty}f'(x)=0$ ?
I can only figure out that $\lim _{x \to -\infty} f(x)$ exists finitely as $f$ is increasing and bounded below .
Please help . Thanks in advance  

Just expanding the answer by David in the comment. Let $g$ be a positive continuous function so that 
$$\int_{-\infty}^0 g(x) <\infty,\ \ \ g(-n) = n\ \ \forall n\in \mathbb N.$$
Then 
$$f(x) = \int_{-\infty}^x g(s) ds$$
satisfies your condition but $f'  = g$ has no limit as $x\to -\infty$. 
An example of such $g$ is 
$$g(x) = \max\{ e^{x}, n(1- n^2| x+n|): n \in \mathbb N\}$$
