Incorrect Result for Multivariable Limit $\lim\limits_{(x,y)\to(0,0)}\frac{2x+y} {x+3y}$

Question

Edit
I just realized that I don't have to worry about this; the worksheet says "Find the value of the limit using Mathematica or justify the limit does not exist". I've already done so by hand, so me asking this is probably pointless.
Thank you to everyone who gave answers, and sorry for wasting your time.

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Welp. I can't find anything helpful, so here we go again.

I'm trying to find the multivariable limit (if it exists) of $$\lim_{(x,y)\to(0,0)}\frac{2x+y} {x+3y}$$ Mathematica says that the limit evaluates to $2$, but I ran it through manually along the x- and y-axes and got that the limit does not exist. Wolfram Alpha agrees with my result.
$f(x,0) = 2; f(0,y) = \frac{1}{3}; 2 \ne \frac{1}{3}$

I have no idea why Mathematica got the result it did or how to fix it, if possible. This was done in Mathematica 11.0, if that makes any difference.

The input and output is as follows:

In:= w[x_, y_] = (2*x + y)/(x + 3*y)
Out= (2*x + y)/(x + 3*y)

In:= Limit[Limit[w[x, y], y -> 0], x -> 0]
Out= 2

By hand:
Along y=0 (x-axis): $f(x,0) = \frac{2x}{x} = 2$
Along x=0: $f(0,y) = \frac{y}{3y} = \frac{1}{3}$
$2 \ne \frac{1}{3}$ so limit does not exist

Answer 1

As with your hand computations,

limYX = Limit[Limit[w[x, y], y -> 0], x -> 0]
2

and

limXY = Limit[Limit[w[x, y], x -> 0], y -> 0]
1/3

To compare with your hand calculations:

limXY == limYX
False

Answer 2

Since version 11.2, Mathematica now has support for genuine multivariate limits. In this case, we can now get the expected answer:

Limit[(2 x + y)/(x + 3 y), {x, y} -> {0, 0}]
   Indeterminate

that is, the limit does not exist, as already demonstrated by other answers.

Answer 3

You can get ANY real result for the function

w[x_, y_] := (2*x + y)/(x + 3*y)

if you let {x,y}->{0,0} in an appropriate manner. Hence the limit does not exist.

Proof in two parts

1) The substitution

w[x, y] /. {y -> t, x -> ((1 - 3 z)/(-2 + z)) t} // Simplify

(* Out[138]= z *)

gives any z except z = 2 (where the substitution breaks down). Notice that the Limit t->0 does not appear explicitly.

2) this substitution gives you

w[x, y] /. {y -> t^2, x -> t} // Simplify

(* Out[140]= (2 + t)/(1 + 3 t) *)

Limit[%, t -> 0]

(* Out[136]= 2 *)

the missing 2.
QED