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How can I show that the limit of the following function, when the variables {x, y} tend to zero, is zero? Mathematica gives the error

{x,y} is not a valid variable.

Domains are $0<x<10$ and $0<y<1$.

Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], {x, y} -> {0, 0}]

Any comment is welcome.

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  • $\begingroup$ Mathematica 12.0 correctly answers Indeterminate (look at the result of Plot3D[Re[ Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]]], {x, -1, 1}, {y, -1, 1}] ). BTW, my ABBY Lingvo x6 says "welcome". Which grammar do you use? $\endgroup$
    – user64494
    Nov 26 '19 at 12:50
  • $\begingroup$ Modern poets do not use capital letters in heads, but your question is not a verse. $\endgroup$
    – user64494
    Nov 26 '19 at 13:05
  • $\begingroup$ @user64494 But if we ContourPlot the function, we see that it tends to zero when the two variables go to zero $\endgroup$
    – user67849
    Nov 26 '19 at 14:47
  • $\begingroup$ Even Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], {x, y} -> {0, 0}, Direction -> "FromAbove"] produces Indeterminate too. $\endgroup$
    – user64494
    Nov 26 '19 at 19:12
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The Limit depends on how you approach {0, 0} so it is Indeterminate. However, you can imply (or specify) a direction of approach which gives 0 as the limit.

$Version

(* "12.0.0 for Mac OS X x86 (64-bit) (April 7, 2019)" *)

Looking at the real part of the function

Plot3D[Re[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]]], {x, -1, 
  1}, {y, -1, 1},
 PlotPoints -> 150,
 MaxRecursion -> 4,
 ClippingStyle -> None,
 AxesLabel -> Automatic]

enter image description here

The multivariate Limit is

Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], 
  {x, y} -> {0, 0}]

(* Indeterminate *)

or

Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], 
  {y, x} -> {0, 0}]

(* Indeterminate *)

A nested Limit is

Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], 
  {x -> 0, y -> 0}]

enter image description here

However, reversing the order of approach

Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], 
  {y -> 0, x -> 0}]

(* 0 *)

Which is the same as

Limit[
 Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], x -> 0],
 y -> 0]

(* 0 *)
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