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.

If the optional argument oterm=false is given, then convert(result,polynom) is applied before returning.

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

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

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

$\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}$

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

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

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

$\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)$

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

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

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

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

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

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

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

The asympt command was updated in Maple 2024.

The oterm option was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.