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Question: How to perform asymptotic limit in Maple for composite variable?

Question:How to perform asymptotic limit in Maple for composite variable?

Ahmed111 135 Maple 2022
infinity asymptotic asympt limit
June 24 2025
0

I'm trying to reproduce a manual asymptotic analysis (see the attached pdf file) in Maple for a two-soliton solution. Specifically, I want to evaluate the limit of a function (e.g., r[2]r[2]r[2] or ∂xq[2]\partial_x q[2]∂x​q[2]). How can I properly perform the limit a2→+−∞ as t​→+−∞ in Maple, either by substitution or by reparametrization, in order to study the asymptotic behavior of a multi-variable expression symbolically? 

> 

restart
> 

with(Student[Calculus1])
> 

lambda1 := I*mu1; lambda2 := I*mu2; a1 := -2*x/mu1+mu1*t; a2 := -2*x/mu2+mu2*t
> 

numer_r := lambda2*cosh(a1)-lambda1*cosh(a2)
> 

denom_r := (lambda1^2+lambda2^2)*cosh(a1)*cosh(a2)-2*lambda1*lambda2*(1+sinh(a1)*sinh(a2))
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r2 := I*(-lambda1^2+lambda2^2)*numer_r/denom_r
> 

numer_dq := (sinh(a1)-sinh(a2))^2; denom_dq := ((lambda1^2+lambda2^2)*cosh(a1)*cosh(a2)-2*lambda1*lambda2*(1+sinh(a1)*sinh(a2)))^2
> 

dq2 := 1-2*(lambda1^2-lambda2^2)^2*numer_dq/denom_dq
> 

limit_r2 := limit(r2, a2 = -infinity); simplify(limit_r2)

Error, invalid input: limit expects its 2nd argument, p, to be of type Or(name = algebraic,set(name = algebraic),list(name = algebraic)), but received -2*x/mu2+mu2*t = -infinity

  

limit_r2

 (1)
> 

limit_r2_pos := limit(r2, a2 = infinity); simplify(limit_r2_pos)

Error, invalid input: limit expects its 2nd argument, p, to be of type Or(name = algebraic,set(name = algebraic),list(name = algebraic)), but received -2*x/mu2+mu2*t = infinity

  

limit_r2_pos

 (2)

NULL

Download asymptotic.mw cooocp_(2).pdf

Any suggestions for reformulating the limit or change of variables would be appreciated. 
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