Verified answer

Note that |x^2 - 9|

= |x - 3| |x + 3|

= |x - 3| |(x - 3) + 6|

≤ |x - 3| [|x - 3| + |6|], by triangle inequality

= |x - 3|^2 + 6 |x - 3|

< |x - 3| + 6|x - 3|, assuming that |x - 3| < 1

= 7|x - 3|.

So given ε > 0, let δ = min {1, ε/7}.

Then, |x - 3| < δ ==> |x^2 - 9| < 7|x - 3| < 7(ε/7) = ε.

I hope this helps!

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