Question

Limit x->1 x*2 - 1 / (x-1)*2

Answer 1

$\rm \lim\limits_{x\to1}\dfrac{x^2-1}{(x-1)^2}\quad=\quad\lim\limits_{x\to1}\dfrac{(x-1)(x+1)}{(x-1)(x-1)}$

The numerator factors into conjugates
while we can also expand the square in the denominator.
Apply a cancellation,

$\rm \lim\limits_{x\to1}\dfrac{x+1}{x-1}$

When you started, you had a function which was approaching this indeterminate form 0/0. But now, the numerator is approaching 2 while the denominator approaches 0.

This is behaving exactly like this limit,

$\rm \lim\limits_{x\to0}\dfrac1x$

The bottom is getting really small,
so the fraction overall is getting huge, approaching infinity.

Oh actually with your problem,
from the right side of 1, the denominator is positive,
so we blow up to infinity.

but from the left side of 1, the denominator is negative,
so we blow up to negative infinity.

So the limit does not exist if it's just x->1.