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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.

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

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  • $\begingroup$ People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful $\endgroup$ – Michael E2 Oct 16 '16 at 21:28
  • $\begingroup$ Related: (21544), (25381) $\endgroup$ – Michael E2 Oct 16 '16 at 21:28
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    $\begingroup$ You're not performing a multivariate limit. You're iterating two single-variable limits. The interior one depends on a generic parameter x and is in fact 2 for all nonzero x. So if the limit exists, it has to be 2. Perhaps the links to the related problems can help you get Mathematica to prove or disprove that it is 2. $\endgroup$ – Michael E2 Oct 16 '16 at 21:31
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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
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    $\begingroup$ I tried Limit[(2 x + y)/(x + 3 y), x | y -> [Infinity]] and got 3/4. $\endgroup$ – Fred Kline Oct 17 '16 at 4:31
  • $\begingroup$ @fred woah, is this even documented syntax? $\endgroup$ – LLlAMnYP Oct 17 '16 at 6:38
  • $\begingroup$ @LLlAMnYP, no. However, we can Reduce to 3/4 and find that whatever is input for x becomes y and vice versa. So, the Limit when $x=y$ is 3/4. $\endgroup$ – Fred Kline Oct 17 '16 at 6:51
  • $\begingroup$ @LLlAMnYP, I use $(a|b)\in\mathbb{N}$ often and just tried it for this problem. $\endgroup$ – Fred Kline Oct 17 '16 at 6:53
  • $\begingroup$ $$\lim_{x|y\to \infty }\frac{2 x+y }{x+3 y}=\frac{3}{4}$$ $\endgroup$ – Fred Kline Oct 17 '16 at 6:59
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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.

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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

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