Asymptotic behavior of a function

mainTerm[expr_, x_] := Module[{approxpoly, prec = 1, j = 0, expon, coef}, approxpoly = Normal[Series[expr, {x, Infinity, prec}]];
	While[approxpoly === 0 && j ≤ 5, j++;
	prec = 2 * prec + j;
	approxpoly = Normal[Series[expr, {x, Infinity, prec}]];];
	approxpoly = PowerExpand@approxpoly;
	expon = Exponent[approxpoly, x];
	coef = Coefficient[approxpoly, x, expon];
	coef * x ^ expon]

mainTerm[(x ^ 3 + 1) / (x + 1), x]

x^2

mainTermAtOrigin[expr_, x_] := mainTerm[expr /. x -> 1 / x, x] /. x -> 1 / x

mainTermAtOrigin[(x ^ 2 + 1) / (x + 1) - 1 / x, x]

-x^(-1)