Limit and intersection point of a function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous and strictly increasing function. Consider the line $y=\alpha x + \beta \; , \; \alpha>0$ . Prove that:

$\lim \limits_{x \rightarrow +\infty} \left ( f(x) - \alpha x - \beta \right ) = -\infty$ .
$\mathcal{C}_f$ has a unique intersection point with the line above.

Solution

As $x \rightarrow +\infty$ we can without loss of generality assume that $x \geq 0$ , hence $f(x) \leq f(0)$ . Thus,
$f(x) - \alpha x - \beta \leq f(0) - \alpha x - \beta = - \alpha x - \gamma \rightarrow - \infty$

Similarly , we can prove that $\lim \limits_{x \rightarrow -\infty} \left ( f(x) - \alpha x - \beta \right ) = +\infty$ .
The function $h(x)=f(x) - \alpha x - \beta \; , \; x \in \mathbb{R}$ is continuous and strictly decreasing. It follows from question (i.) that $h \left ( \mathbb{R} \right ) = \mathbb{R}$ . Since $0 \in \mathbb{R}$ it follows that there exists an $x_0 \in \mathbb{R}$ such that $h(x_0)=0$ which is unique due to the monotony of $h$ .