Limit of ((x^2)/(x+1))-((x^2+2)/(x+3)) in inifinity returns -infinity instead of 2

[SomebodyLikeEveryBody]

If we do

limit(((2*x^2-2*x-2)/(x^2+4*x+3)), x, Infinity)

It returns -Infinity instead of 2,

and when we divide by x^2, it works

limit(((2-((2)/(x))-((2)/(x^2)))/(1+((4)/(x))+((3)/(x^2)))), x, Infinity)

==> returns 2