# Limit of a two variable function at the origin

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 $$0 and $$0.

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


Any comment is welcome.

• 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? Nov 26 '19 at 12:50
• Modern poets do not use capital letters in heads, but your question is not a verse. Nov 26 '19 at 13:05
• @user64494 But if we ContourPlot the function, we see that it tends to zero when the two variables go to zero
– user67849
Nov 26 '19 at 14:47
• Even Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]], {x, y} -> {0, 0}, Direction -> "FromAbove"] produces Indeterminate too. Nov 26 '19 at 19:12

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] 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}] 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 *)