Mathematic correct plotting rational function What is range of $$-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2}$$
assuming $\epsilon\in(0,1)$?
It looks like $$\lim_{\epsilon\rightarrow1}-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2}=\lim_{\epsilon\rightarrow1}-\frac{-4+6\epsilon}{-2\epsilon}=1$$
However plotting on mathematica $-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2}$ explodes to infinity as $\epsilon\rightarrow1$.
Is $2$ an upper bound for $$1-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2}$$ with $\epsilon\in(0,1)$?

There is no problem in Mathematica!
If you define
$$T=1-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2},$$
And use the following command:
T//Simplify,

the output generated is
$$\frac{4\epsilon}{1+\epsilon}$$
Mathematica will, in general, not automatically simplify your expression.
