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Is it possible to give Mathematica a limit about's function at infinity? For example I would like to

Limit[F[r], r -> \[Infinity], Assumptions -> F[\[Infinity]] == 0]

To evaluate to zero. What are my options? And will is there a way to impose stronger conditions, for example if $F(x)$ would behave as $1/r^2$ at infinity is it possible to assume this in Mathematica? So that for example one might calculate

$$ \lim_{r \to \infty} r F(r) = 0 \quad \text{for} \; F(r) \approx 1/r^2 \; \text{with} \; r \gg1. $$

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  • $\begingroup$ What would you do with this if it were possible? What do you need this functionality for? Maybe there's a different approach to your actual underlying problem. $\endgroup$
    – MarcoB
    Commented Mar 18, 2023 at 13:40
  • $\begingroup$ Finding a Green's function for a ODE.. I wanted to check that the Green's function that I found is really a Green's function. I.e. $L \int G(x,y) F(y) dy = F(x)$ with $L$ being some differential operator (acting on $x$). Since the expression is quite complicated I choose the upper limit as integral to be some value instead of $\infty$ to "help" Mathematica, so now I'm finding the limit of that expression. $\endgroup$
    – Nitaa a
    Commented Mar 18, 2023 at 13:50

1 Answer 1

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ClearAll[a, F, r, t, x]

Use TagSet to define UpValues for F

F /: Asymptotic[F[x_], x_ -> Infinity] = 1/x^2;

F /: Limit[F[x_], x_ -> Infinity] = 0;

UpValues[F]

enter image description here

Asymptotic[F[r], r -> Infinity]

(* 1/r^2 *)

Limit[F[r], r -> Infinity]

(* 0 *)

F cannot be buried any deeper in the LHS of the TagSet. Use a replacement rule when F is buried deeper.

asymFinf = F[t_] -> Asymptotic[F[t], t -> Infinity]

(* F[t_] -> 1/t^2 *)

Asymptotic[a*F[r] /. asymFinf, r -> Infinity]

(* a/r^2 *)

Limit[a*F[r] /. asymFinf, r -> Infinity]

(* 0 *)

Limit[r*F[r] /. asymFinf, r -> Infinity]

(* 0 *)

Limit[a*r^2*F[r] /. asymFinf, r -> Infinity]

(* a *)
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