# Demonstration analysis and good books?

by fgyamauti
Tags: analysis, books, demonstration
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 You want $|x^2- 4|= |(x- 2)(x+ 2)|= |x+ 2||x- 2|< \epsilon$. So we can write $|x- 2|< \epsilon/|x+ 2|$ But we need a value, $\delta$ that does NOT depend on x so that is $|x- 2|< \delta$ then $|x- 2|< \epsilon/|x+ 2|$. That means we want $\delta< \epsilon/|x+2|$. If |x- 2|< 1 (The "1" is chosen just because it is simple. Any positive number would do.) then -1< x- 2< 1 so 1< x< 3 and then 3< x+2< 5 so that 3< |x+ 2|< 5. Then $$\frac{1}{5}< \frac{1}{|x+2|}< 1/3$$ and so $$\frac{\epsilon}{5}< \frac{\epsilon}{|x+2|}$$ That is, if $$|x- 2|< \frac{\epsilon}{5}$$ and |x- 2|< 1 as assumed above to get this, we will have $$|x-2|< \frac{\epsilon}{|x+2|}$$ so $$|x-2||x+2|= |x^2- 4|< \epsilon$$ as need. In other words, take $\delta$ to be the smaller of $\epsilon/5$ or 1. Analysis is basically the theory behind the Calculus. I would NOT take an analysis course without having completed the Calculus sequence.