bwpbruce
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Given:
[math]\lim_{x \to 3} \dfrac{2}{x + 3} = \dfrac{1}{3}[/math]
Goal:
Prove using $\epsilon-\delta$ definition of a limit.
Plan:
Let [math] \epsilon > 0 [/math] be given. Find [math] \delta > 0 [/math] so that
if [math] 0 < \left|x - 3 \right| < \delta [/math], then [math] \left|f(x) - \dfrac{1}{3} \right| < \epsilon [/math]
Solve:
[math] \left|\dfrac{2}{x + 3} - \dfrac{1}{3} \right| = \left|\dfrac{6}{3(x + 3)} - \dfrac{x + 3}{3(x + 3)} \right| = \left| \dfrac{x + 3}{3(x + 3)} - \dfrac{6}{3(x + 3)}\right| = \left| \dfrac{x + 3 - 6}{3(x + 3)}\right|= \left| \dfrac{x -3}{3(x + 3)}\right| < \epsilon [/math]
\begin{align*} |x - 3| &< |3(x + 3)|\epsilon \\ x\text{ is approaching }3, \text{therefore }|x - 3| &< |3(3 + 3)|\epsilon \\ |x - 3| &< |3(6)| \\ \left|x - 3 \right| &< 18\epsilon\end{align*}
Result:
[math] 18\epsilon = \delta [/math]
Conclusion:
From the statement [math]18\epsilon = \delta [/math], it is clear that for any given [math]\epsilon[/math], a [math]\delta[/math] exists such that if [math] 0 < \left|x - 3 \right| < \delta [/math], then [math] \left|f(x) - \dfrac{1}{3} \right| < \epsilon [/math]
[math]\lim_{x \to 3} \dfrac{2}{x + 3} = \dfrac{1}{3}[/math]
Goal:
Prove using $\epsilon-\delta$ definition of a limit.
Plan:
Let [math] \epsilon > 0 [/math] be given. Find [math] \delta > 0 [/math] so that
if [math] 0 < \left|x - 3 \right| < \delta [/math], then [math] \left|f(x) - \dfrac{1}{3} \right| < \epsilon [/math]
Solve:
[math] \left|\dfrac{2}{x + 3} - \dfrac{1}{3} \right| = \left|\dfrac{6}{3(x + 3)} - \dfrac{x + 3}{3(x + 3)} \right| = \left| \dfrac{x + 3}{3(x + 3)} - \dfrac{6}{3(x + 3)}\right| = \left| \dfrac{x + 3 - 6}{3(x + 3)}\right|= \left| \dfrac{x -3}{3(x + 3)}\right| < \epsilon [/math]
\begin{align*} |x - 3| &< |3(x + 3)|\epsilon \\ x\text{ is approaching }3, \text{therefore }|x - 3| &< |3(3 + 3)|\epsilon \\ |x - 3| &< |3(6)| \\ \left|x - 3 \right| &< 18\epsilon\end{align*}
Result:
[math] 18\epsilon = \delta [/math]
Conclusion:
From the statement [math]18\epsilon = \delta [/math], it is clear that for any given [math]\epsilon[/math], a [math]\delta[/math] exists such that if [math] 0 < \left|x - 3 \right| < \delta [/math], then [math] \left|f(x) - \dfrac{1}{3} \right| < \epsilon [/math]