Answer:
You are technically dividing by zero, it breaks math.
In the step (X+1)(X-1)=X-1 → X+1=1.
You divide (X+1)(X-1)=X-1 by (x-1) on both sides to get:
(x+1)(x-1) / (x-1) = (x-1) / (x-1).
Since x = 1,
(x + 1)(x - 1) / (x - 1) = (x - 1) / (x - 1)
becomes (2)(0) / 0 = 0 / 0.
And when you divide by zero, it's undefined since you cannot multiply a number by zero and get a number other than zero, it's undefinable.
This is where limits come in, when you divide by a variable, you have to assumed that is it ≠ 0.
You know that 1 ≠ 2, and 0 ≠ 1, and so on.
As dividing by zero approaches infinity,
1/0.1 = 10, 1/0.01 = 100, 1/0.001 = 1000, 1/0.00..1 = 1000.. → 1/0 → infinity.
1/-0.1 = -10, 1/-0.01 = -100, 1/-0.001 = -1000,
1/-0.00..1 = -1000.. → 1/0 → -infinity.
This suggests that dividing by zero should be infinity.
Therefore infinity = -infinity, Wrong.
When you are dividing, you are really asking yourself, how many times does this number fit into the other number.
1 cannot fit into 0, there isn't anything to fit into. This can also be thought of as:
1 = 0 + 0 + 0 + 0 + 0 + ......
0 + 0 = 0, so 1 ≠ 0.