Example 4 steps, run in order in one session. Every step was executed against the engine and the output below is what it produced.

Step 1

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]

No output - this step sets something up for the next one.

Step 2

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

Output

x^2

Step 3

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

No output - this step sets something up for the next one.

Step 4

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

Output

-x^(-1)

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Asymptotic behavior of a function

Functions used

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