Proving Epsilon-Delta Inequalities for Limits of Functions

[Turtle1991]

Homework Statement

Show that, if $$0<\delta<1$$ and $$|x-3|<\delta$$, then $$|x^{2}-9|<7\delta$$

Homework Equations

The Attempt at a Solution

I know that I need to transform one of the inequalities into the form of the other to prove it, but I don't see how. I can plug in values and of course it works since the limit of x^2 as x approaches 3 is 9, but I don't see how to show it algebraically. Please help

 

[quasar987]

|x²-9| = |x+3||x-3|<|x+3|d

From there, how can you use the triangle inequality and the hypothesis of |x-3|<d and 0<d<1 to get the result?

 

[HallsofIvy]

Note that if |x-3|< $\delta$ and and $\delta$< 1, then |x-3|< 1 so -1< x-3< 1. What does that tell you about x+ 3 and |x+3|?