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I had the following question of a student this day about proving the following limit:

[tex]

\lim_{x \rightarrow 3} x^2 = 9

[/tex]

So this means that I should prove that

[tex]

|x-3| < \delta \ \rightarrow \ \ |x^2 - 9| < \epsilon(\delta)

[/tex]

So I had the following idea:

[tex]

|x^2 - 9| = |x-3||x+3|

[/tex]

The first term on the RHS is smaller than delta. For the second term I write

[tex]

|x+3| = |x-3+6| < |x-3| + 6

[/tex]

So I get in total

[tex]

|x-3| < \delta \ \rightarrow \ |x^2 - 9| = |x-3||x+3| < |x-3|(|x-3| + 6 ) < \delta(\delta + 6)

[/tex]

So choosing

[tex]

\epsilon = \delta(\delta + 6)

[/tex]

should prove the statement, right? But if I see the book they're using (Apostol) I see that they "choose" |x-3|<C, try to make an inequality then for |x+3| etc. But for me that seems making things more difficult then they are. So my question is: is my answer described above right?