The function asympt computes the asymptotic expansion of f with respect to the variable x (as x approaches infinity).

The third argument n specifies the truncation order of the series expansion.  If no third argument is given, the value of the global variable Order (default Order = 6) is used.

Specifically, asympt is defined in terms of the series function

However, the expression returned will be in sum-of-products form rather than in the series form.

$\mathrm{asympt}\left(\frac{x}{1-x-x^2}\,x\right)$

$-\frac{1}{x}+\frac{1}{x^2}-\frac{2}{x^3}+\frac{3}{x^4}-\frac{5}{x^5}+\mathrm{O}\left(\frac{1}{x^6}\right)$

$\mathrm{asympt}\left(\frac{n\!}{\sqrt{2\mathrm{Pi}}}\,n\,3\right)$

$\frac{\frac{1}{\sqrt{\frac{1}{n}}}+\frac{\sqrt{\frac{1}{n}}}{12}+\frac{{\left(\frac{1}{n}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{288}+\mathrm{O}\left({\left(\frac{1}{n}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}{{\left(\frac{1}{n}\right)}^n{\ⅇ}^n}$

$\mathrm{asympt}\left({\ⅇ}^{x^2}\left(1-\mathrm{erf}\left(x\right)\right)\,x\right)$

$\frac{1}{\sqrt{\mathrm{\pi }}x}-\frac{1}{2\sqrt{\mathrm{\pi }}{x}^3}+\frac{3}{4\sqrt{\mathrm{\pi }}{x}^5}+\mathrm{O}\left(\frac{1}{x^7}\right)$

$\mathrm{asympt}\left(\sqrt{\frac{\mathrm{Pi}}{2}}\mathrm{BesselJ}\left(0\,x\right)\,x\,3\right)$

$\sin \left(x+\frac{\mathrm{\pi }}{4}\right)\sqrt{\frac{1}{x}}-\frac{\cos \left(x+\frac{\mathrm{\pi }}{4}\right){\left(\frac{1}{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{8}-\frac{9\sin \left(x+\frac{\mathrm{\pi }}{4}\right){\left(\frac{1}{x}\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{128}+\mathrm{O}\left({\left(\frac{1}{x}\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

$\mathrm{series}\left(\frac{\ln \left(x\right)}{x-1}\,x\,8\right)$

$-\ln \left(x\right)-\ln \left(x\right)x-\ln \left(x\right)x^2-\ln \left(x\right)x^3-\ln \left(x\right)x^4-\ln \left(x\right)x^5-\ln \left(x\right)x^6-\ln \left(x\right)x^7+O\left(x^8\right)$

$\mathrm{asympt}\left(\frac{\ln \left(x\right)}{x-1}\,x\,8\right)$

$\frac{\ln \left(x\right)}{x}+\frac{\ln \left(x\right)}{x^2}+\frac{\ln \left(x\right)}{x^3}+\frac{\ln \left(x\right)}{x^4}+\frac{\ln \left(x\right)}{x^5}+\frac{\ln \left(x\right)}{x^6}+\frac{\ln \left(x\right)}{x^7}+\mathrm{O}\left(\frac{1}{x^8}\right)$

The asympt command was updated in Maple 2016; see Advanced Math.