**Question**: I kinda forgot my multivariable limits already...

Is it legit to put a condition under a limit eg $\lim\_{\substack{(x,y) \to (0,0) }} \frac{x}{x^2+y^2}$ doesn't exist because

$$\lim\_{\substack{(x,y) \to (0,0) \\ y=0 }} \frac{x}{x^2+y^2}$$

$$= \lim\_{\substack{(x,y) \to (0,0) \\ y=0 }} \frac{x}{x^2}$$

$$= \lim\_{\substack{(x,y) \to (0,0)}} \frac{x}{x^2}$$

$$= \lim\_{\substack{x \to 0}} \frac{x}{x^2}$$

$$= \lim\_{\substack{x \to 0}} \frac{1}{x}$$

doesn't exist

?

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**Explanation**: The thing is in single variable real: it's legit to say $\lim\_{x \to 0^+} \frac 1 x$ because there's a precise definition for this where instead of $- \delta < x - 0 < \delta$ you say $0 < x - 0 < \delta$ or something. Hence, here it's legit to write $\lim\_{\substack{x \to 0 \\ x > 0}} \frac 1 x := \lim\_{x \to 0^+} \frac 1 x$.

I would like to know this is similarly legit in complex/real multivariable, if it is. I think if this isn't legit, then I would like to know the proper way to argue. Perhaps say $\frac{x}{x^2+y^2} = \frac{1}{x}$ for $y=0$ and thus... (wait let's change to $x=0$) let's say $\frac{x}{x^2+y^2} = 0$ for $x=0$ and thus $\frac{x}{x^2+y^2} \to 0$ as $y \to 0$ along $x=0$...but I don't know how to make this precise either.

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(Context: I'm actually asking with complex limits in mind, but I guess it will be [equivalent to real multivariable limits anyway](https://math.stackexchange.com/questions/4268942/what-exactly-are-the-differences-between-real-multivariable-limits-and-complex-l).)

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**Guess**:

Ah wait writing all this I think I figured it out.

(convert to answer)