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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:

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

Solution

  1. 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.

  2. 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.

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