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The other day I was asked to solve the following multivariable limit:

$$\lim_{(x,y)\to(0,0)}\frac{x-y}{x+y}$$

As simple as it was I was stumped. I tried some algebraic manipulation using conjugates, which got me nowhere. Then I tried defining $x$ and $y$ by their polar definitions and got stuck here:

$$\lim_{r\to0}\frac{r\cos\theta-r\sin\theta}{r\cos\theta+r\sin\theta}$$

$$=\frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}$$

But I didn't know what to do from here.

The only other approach that struck me as a possibility was partial fractions.

What's the best way to tackle this limits problem and ones like it?

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    Take for example $y=x/n$ and see what happens. – 2017-02-19
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    You look for different ways to approach the origin along continuous curves $(x,g(x)),$ where $g(x) \to 0,$ or $(f(y),y)$ where $f(y) \to 0,$ etc. If you'll find a couple of such curves such that the corresponding limits, say $\lim_{x \to 0} F(x,g_1(x))$ and $\lim_{x \to 0} F(x,g_2(x))$ are not equal, then the limit in question does not exist. – 2017-02-19
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    Just approaching along the two axes should be enough. – 2017-02-19
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    ...in THIS case, yeah. Knowledge of a general principle is always better. – 2017-02-19

3 Answers 3

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This limit does not exist. If you approach $(0,0)$ through the $x$-axis, you get

$$\lim_{(x,y)\to(0,0)}\frac{x-y}{x+y}=\lim_{x\to0}\frac{x}{x}=1,$$

But if you approach it by the line $y=x$, you get

$$\lim_{(x,y)\to(0,0)}\frac{x-y}{x+y}=\lim_{x\to0}\frac{x-x}{x+x}=0,$$

So the limit can't exist.

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Your “polar coordinates” method can work: $$\lim_{r\to0}\frac{r\cos\theta-r\sin\theta}{r\cos\theta+r\sin\theta}=\frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}$$ If the two-variable limit existed, the expression on the right would not depend on $\theta$. But clearly it does depend on $\theta$. If $\theta = 0$, the limit is $1$. But if $\theta = \frac{\pi}{2}$, the limit is $-1$.

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Take the next sequences:

$x_k=\left\{\displaystyle\frac{1}{k},\displaystyle\frac{1}{k} \right\}$. Clearly, $\lim\limits_{k\to\infty}x_k=(0,0)$. And, $\lim\limits_{k\to\infty}f(x_k)=0$

$y_k=\left\{ \displaystyle\frac{1}{k},0\right\}$. Clearly, $\lim\limits_{k\to\infty}y_k=(0,0)$ and $\lim\limits_{k\to\infty}f(y_k)=1$

We can conclude that the limit doesn't exist. Why? We take two sequences such that converges to $(0,0)$ but the sequence of images converges to two different values.