# Chapter 2 - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises - Page 109: 96

Prove that for every negative real number $-B$, there exists a corresponding $\delta\gt0$ such that for all $x$ $$2-\delta\lt x\lt 2\Rightarrow \frac{1}{x-2}\lt -B$$

#### Work Step by Step

$$\lim_{x\to2^-}\frac{1}{x-2}=-\infty$$ According to the formal definition deduced from Exercise 93, we need to prove that for every negative real number $-B$, there exists a corresponding $\delta\gt0$ such that for all $x$ $$2-\delta\lt x\lt 2\Rightarrow \frac{1}{x-2}\lt -B$$ - Examine the inequality: $$\frac{1}{x-2}\lt -B$$ $$x-2\gt-\frac{1}{B}$$ $$x\gt2-\frac{1}{B}$$ - So if we set $\delta=\frac{1}{B}$, then we have $2-\frac{1}{B}\lt x\lt 2$, meaning that we would for all $x$ have $\frac{1}{x-2}\lt -B$, satisfying the requirements. The proof has been completed.

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