Limiting form of a polynomial expression

Question

When simplifying an expression by hand, a trick that is often used is to remove terms that are lower powers of the independent variable, for instance, as $x \rightarrow \infty$,

$x^2 + x$

becomes simply

$x^2$

I am trying to replicate this functionality in Mathematica. Limit seems like an obvious choice, but

Limit[x^2 + x, x -> ∞]

simply returns $∞$ (not terribly helpful). Another option is Series, but it seems that one has to pass the highest power in the expression to get the desired result, with

Normal@Series[x^2 + x, {x, ∞, -2}]

returning the expected $x^2$. However, this does not work in general for more complicated expressions, when the highest power may not be so obvious. Consider the following example:

$\frac{x^2+x}{x+1}$

Mathematica code

(x^2+x)/(x+1)

We expect that the limiting form of this expression as $x \rightarrow \infty$ should be $x$ (the top becomes $x^2$, the bottom becomes $x$, and this then simplifies to $x$). However, if you just use the highest power available, Series returns 0 (you have to actually use Normal@Series[x^2 + x, {x, ∞, -1}]).

My question is, can anyone produce a generalized piece of code that can produce a limiting expression? It would be good if the procedure handled the other extreme as well, when the variable in question approaches 0 (in which case $x^2 + x$ would simplify to $x$).

Edit: It seems that the main issue for the solutions presented so far is that the form of the expression is not consistent. These operations are relatively easy to perform by hand, but sometimes tedious, which is why it would be nice to have Mathematica do it for me! Here's a pathological example that I can reduce by hand and for which the accepted solution should be able to return the correct result:

$\frac{\sqrt{x^3 + x}}{x^5 + x + 1} \frac{x^2}{x^{10} + x} + \frac{x^5 + x}{x}$

Mathematica code

Sqrt[x^3 + x]/(x^5 + x + 1) x^2/(x^10 + x) + (x^5 + x)/x

(The right answer is $x^4$)

Answer 1

Method I

(1) Take series at infinity to get a (Laurent) polynomial.

(2) Find largest exponent.

(3) Find corresponding coefficient.

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
  ]

Example:

In[83]:= mainTerm[(x^3 + 1)/(x + 1), x]

(* Out[83]= x^2 *)

Method II

(1) Take the series at infinity to get a Laurent polynomial.

(2) Homogenize it wrt a new variable.

(3) Set that new variable to zero.

homogenize[poly_, vars_, new_] := 
 Module[{degfunc, totdeg, j}, 
  degfunc = Apply[Plus, Map[Function[x, Exponent[#, x] &], vars]];
  degfunc = Distribute[degfunc, Function, Plus];
  totdeg = Max[Map[degfunc, Apply[List, poly]]];
  Apply[Plus, 
   Table[poly[[j]]*new^(totdeg - degfunc[poly[[j]]]), {j, 
     Length[poly]}]]]

mainTerm2[expr_, x_] := Module[
  {approxpoly, prec = 1, j = 0, hpoly, y},
  approxpoly = Normal[Series[expr, {x, Infinity, prec}]];
  While[approxpoly === 0 && j <= 5,
   j++;
   prec = 2*prec + j;
   approxpoly = Normal[Series[expr, {x, Infinity, prec}]];
   ];
  hpoly = homogenize[approxpoly, {x}, y];
  hpoly /. y -> 0
  ]

Example:

In[91]:= mainTerm2[(x^3 + 1)/(x + 1), x]

(* Out[91]= x^2 *)

--- edit ---

For getting the asymptotic behavior at the origin you can do as follows.

(1) Replace x by 1/x. (2) Get the asymptotic term at infinity. (3) Transform back with x->1/x.

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

Example:

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

(* Out[106]= -(1/x) *)

--- end edit ---

--- edit 2 ---

After adding a PowerExpand to deal appropriately with fractional power issues, we can wotk with Puisieux expansions. Here is an example.

ee = Sqrt[x^3 + x]/(x^5 + x + 1)*x^2/(x^10 + x^7) + (x^5 + x)/x;

In[170]:= mt = mainTermAtOrigin[ee, x]

(* Out[170]= (1/x)^(9/2) *)

We can test that this is correct by checking whether the expression divided by it approaches unity as x goes to zero.

In[171]:= Limit[ee/mt, x -> 0]

(* Out[171]= 1 *)

Answer 2

The solution to this requires just a minor modification of my answer here:

ClearAll[limitingTerm1]
SetAttributes[limitingTerm1, HoldAll]
limitingTerm1[expr_, var_Symbol, func_] := var^Exponent[#, var, func] &@Simplify[expr]

This gives the right answer for both the examples provided by the OP:

limitingTerm1[(x^2 + x)/(x + 1), x, Max]
(* x *)

limitingTerm1[Sqrt[x^3 + x]/(x^5 + x + 1) x^2/(x^10 + x) + (x^5 + x)/x, x, Max]
(* x^4 *)

---

One can now generalize this to polynomials with arbitrary coefficients. It works as a two-pass approach, where the second pass is applied to the result of the first if its denominator still contains an x term. This seems to work very well in examples I've tested, but it might be possible to find edge cases where it doesn't work. Nevertheless, some modifications and extensions along these lines should help solve it.

ClearAll[limitingTerm2]
SetAttributes[limitingTerm2, HoldAll]
limitingTerm2[expr_, var_Symbol, func_] := Module[{term},
    term := With[{pow = Exponent[#, var, func]}, Coefficient[#, var, pow] var^pow] &;
    If[FreeQ[Denominator@#, x], #, term@Apart@#] &@term@Simplify[expr]
]

limitingTerm2[(a x^2 + bx)/(d + c x), x, Max]
(* (a x)/c *)

It also gives the correct answer for the other examples above.

Answer 3

For finding the leading term of an asymptotic expansion, you can use the answer I gave to Get leading series expansion term?:

leadingSeries[expr_, {x_, x0_}] := Normal[
    expr /. 
        x -> Series[x, {x, x0, 1}] /.
        Verbatim[SeriesData][a__, {b_, ___}, c__] :> SeriesData[a, {b}, c]
]

For your examples:

leadingSeries[x^2+x, {x, Infinity}]
leadingSeries[(x^2+x)/(x+1), {x, Infinity}]
leadingSeries[Sqrt[x^3+x]/(x^5+x+1) x^2/(x^10+x)+(x^5+x)/x, {x, Infinity}]

x^2

x

x^4

The accepted answer gives an incorrect result for something like:

expr = 1 + x + Sqrt[x^5]/(x+1);

limitingTerm1[expr, x, Max]
leadingSeries[expr, {x, Infinity}]

x

x^(3/2)

Answer 4

This works for me:

Normal@Series[((x^2 + x ) / (x + 1) ) , {x, Infinity, 1}] -> x

or this..

Normal@Series[((x^2 + x ) / (x + 1) /. x -> 1/y ) , {y, 0, 1}] /.  y -> 1/x

Note this example is a case where there is not a Taylor series, but Series[] is being smart for you and returning a negative power solution, which should give a clue why a fully general method is going to be problematic.

Edit another approach:

n = 0;
While[ (lim = Limit[(1/x)^n (x^3 + x ) / (x + 1)  , x -> Infinity]) ==  Infinity , n++];
lim x^(n)

Obvoulsly you'd want to limit n to prevent runaway.

I'm suspecting your question isn't mathematically well posed. Suppse the "limit" is a Log[x], do you expect your 'general' approach to pick that up?