Proving $\lim _{ x\to\infty }{ f(x) }=\infty$ if $f'(x)>c$ for every $x$

Question

Given a differentiable function $f: (0,\infty) \rightarrow \mathbb R $ and $c>0$ such that $f'(x)>c$ for every $x$.

Prove: $\lim _{ x\rightarrow\infty }{ f(x) }=\infty$

Using the MVT, I got to $f(x)>c(x-x_0)+f(x_0)$, and I think I should proceed using the definition of limit, but I got stuck.

Any help appreciated.

Answer 1

For $x>1$, we have that $f(x)-f(1)=\int_1^xf'(t)\,dt>c(x-1)$. Hence,

$$f(x)>c(x-1)+f(1)$$

And it's easy to see that the RHS is unbounded as $x\to\infty$.

EDIT: As @zhw. noted, the solution above does not work if $f'$ is not Riemann integrable. Nonetheless, we may use the general idea to craft a solution that does not involve integration.

Consider $g(x)=c(x-1)+f(1)$. Then $h(x)=f(x)-g(x)$ is such that $h(1)=0$ and $h'(x)>0$ for all $x>1$. It follows that $h(x)>0$ for all $x>1$, that is,

$$f(x)>c(x-1)+f(1)$$

for all $x>1$. The conclusion follows.