# Approximation of $f(x)$ at large values of $x$?

Suppose you a function $f(x)=1/x+x^2$. At large values of $x$, it is closely approximated by $f(x)=x^2$.

Can Mathematica do this for me for a more complicated function? How would I input the command?

Take a series around infinity, dropping all $1/x^n$ terms.

Series[1/x + x^2, {x, \[Infinity], 0}]
(* x^2+O[1/x]^1 *)


Use Normal to obtain an ordinary expression from the SeriesData object.

A slightly more complicated example: the hyperbola represented by $\sqrt{x^2-1}$ has asymptote $y=x$ for positive $x$; so we get:

Normal@Series[Sqrt[x^2 - 1], {x, \[Infinity], 0}]
(* x *)


If you want to keep the function as it is without converting to series than you may try this:

1/x + x^2 /. (y_ /; (Limit[y, x -> \[Infinity]] == 0) :> 0)
(*x^2*)
Sin[x] + 1/x /. (y_ /; (Limit[y, x -> \[Infinity]] == 0) :> 0)
(*Sin[x]*)


You may also find this link useful (here)

• Note that this fails in cases like Sqrt[x^2-1] or x^2/(1+x^2), resulting in no simplification, and zero, respectively, i.e. It only handles cases where the decaying terms are totally separated already. – 2012rcampion Feb 10 '15 at 20:34