1. ## Limits

This is probably pretty easy and I'm just over-thinking it, but I have no clue what to do for this problem.

If $\displaystyle \lim_{x \to3} (3x-2)=7$, find $\displaystyle S$ such that $\displaystyle |(3x-2)-7|<0.003$ whenever $\displaystyle 0<|x-3|<S.$

2. Originally Posted by summermagic
This is probably pretty easy and I'm just over-thinking it, but I have no clue what to do for this problem.

If $\displaystyle \lim_{x \to3} (3x-2)=7$, find $\displaystyle S$ such that $\displaystyle |(3x-2)-7|<0.003$ whenever $\displaystyle 0<|x-3|<S.$
Note that $\displaystyle \left|(3x-2)-7\right|=\left|3x-9\right|=3{\color{red}\left|x-3\right|}<0.003$ whenever $\displaystyle 0<{\color{red}\left|x-3\right|}<S.$

With this, what is S?

3. I think S is just a symbol for whatever that number is that is supposed to fit in the last equation.

But so therefore, S = 0.001?

4. Originally Posted by summermagic
But so therefore, S = 0.001?