Can someone explain what I circled in red here? This is my teacher's work. Why was $x + 1$ replaced and can someone explain the $1 \lt |x-1| \lt 3$ to me? I am really confused and I would like some "baby step" answers.
[Math] Epsilon-Delta limit proof
epsilon-delta
Best Answer
Since you are working with $x \to -1$, you can always assume that $x$ is as close as you need to $-1$. In particular, you can restrict to those values of $x$ such that $x$ lies in an interval of radius $1$ centered at $-1$, and therefore $-2<x<0$. Then $-2-1<x-1<0-1$, i.e. $-3<x-1<-1$, or $1<|x-1|<3$. Now you can say that $3|x+1||x-1|<3 \cdot 3 \cdot |x+1|$.