Limit of (x^2)/(x^2-1) when x->infinity

According to the answer sheet it becomes

limit of (1/x^2)/(1+1/x^2) when x->infinity

which in turn equals 1. My question is, what happens to turn the original equation into the new one?

If anyone could help me understand the process I'd be very grateful.

How about dividing both numerator and denominator by x² ?

But either the first expression is incorrect or the second on is, since you start off with subtraction in denominator, and end up with sum in denominator. Either way, you get same result:

lim[x→∞] x² / (x² − 1)
= lim[x→∞] {x²/x²} / {(x² − 1)/x²}
= lim[x→∞] 1 / (1 − 1/x²)
= 1 / (1 − 0)
= 1

lim[x→∞] x² / (x² + 1)
= lim[x→∞] {x²/x²} / {(x² + 1)/x²}
= lim[x→∞] 1 / (1 + 1/x²)
= 1 / (1 + 0)
= 1

There must be a typo.
lim_{x→∞} (1/(x^2)) / (1 + 1/(x^2))
= lim_{x→∞} (1/(x^2)) / ((x^2 + 1)/(x^2))
= lim_{x→∞} ((x^2)/(x^2)) / (x^2 + 1)
= lim_{x→∞} 1 / (x^2 + 1)
= 0.

(x^2) / (x^2 - 1) =>
(x^2 + 1 - 1) / (x^2 - 1) =>
(x^2 - 1) / (x^2 - 1) + 1 / (x^2 - 1) =>
1 + 1 / (x^2 - 1)

x goes to infinity

1 + 1 / (inf) =>
1 + 0 =>
1