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Problem 71

Graphical Reasoning The statement

$$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4$$

means that for each $\varepsilon>0$ there corresponds a $\delta>0$ such that if $0<|x-2|<\delta,$ then

$$\left|\frac{x^{2}-4}{x-2}-4\right|<\varepsilon$$

If $\varepsilon=0.001,$ then
$$\left|\frac{x^{2}-4}{x-2}-4\right|<0.001$$

Use a graphing utility to graph each side of this inequality. Use the zoom feature to find an interval $(2-\delta, 2+\delta)$ such that the inequality is true.

$(1.9995,2.0005)$

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