Prove: $\lim_{x \to \infty} f'(x) = 2$ indicate $\lim_{x \to \infty}f(x) = \infty$ This question is from last year exam in calculus.
In the answers they mentioned more than one way to solve this question.
The method they used is to build a new function but it was a bit complicated.
I would like to get some help.

The key is to show that $f$ is bounded below by a function which goes to infinity as $x \to \infty$.  It follows from your hypothesis that there exists an $x_0$ such that for all $x \geq x_0$, $f'(x) \geq 1$.
Now let $x$ be any number bigger than $x_0$.  Consider the average rate of change
$$\frac{f(x) - f(x_0)}{x-x_0}$$
between $x$ and $x_0$.  By the mean value theorem, there exists a $c > x_0$ such that
$$\frac{f(x) - f(x_0)}{x-x_0} = f'(c)$$
Since $f'(c) \geq 1$, this implies that there is a constant $K$ such that $f(x) \geq x +K$ for all $x > x_0$.
