some limiting information about a function from it's first and second derivative

Question

Let $f$ be a twice differentiable function on $\mathbb R $ such that both $f^{'}$ and $f^{''}$ are strictly positive on $\mathbb R $. Then is there a functon such that $ \lim_{x\to \infty} f(x) \neq \infty$ ?

I can intuitively imagine the types of functions on $\mathbb R $ such that both $f^{'}$ and $f^{''}$ are strictly positive on $\mathbb R $ which grows very slowly and converges to some finite number.But i am not able to find an explicit function or a proof.

Answer 1

Consider $f(x)=e^{-x}$ as a counterexample