How to Interpret Multidimensional Limits?

[melknin]

I'm trying to understand how to interpret multidemensional limits. For example, suppose you have the following:

$$\lim\limits_{x \to \infty}\lim\limits_{y \to \infty} x\frac{1}{y}$$

Would this be infinity, 0, or 1?

This is really a more general version of the question I'm working with regarding the behavior of a function that has the property $$f_a(b) \to\limits_{a \to \infty} 1$$ and $$f_a(b) \to\limits_{b \to \infty} \infty$$ in the context that both a and b are going to infinity.

Thanks in advance for any help!

 

[Galileo]

I'm trying to understand how to interpret multidemensional limits. For example, suppose you have the following:

$$\lim\limits_{x \to \infty}\lim\limits_{y \to \infty} x\frac{1}{y}$$

Would this be infinity, 0, or 1?

Well,
$$\lim \limits_{x \to \infty}\left(\lim\limits_{y \to \infty} \frac{x}{y}\right)=\lim \limits_{x \to \infty} 0=0$$
,but
$$\lim\limits_{y \to \infty}\left(\lim\limits_{x \to \infty} \frac{x}{y}\right )$$
doesn't exist, because $\lim\limits_{x \to \infty} \frac{x}{y}$ doesn't exist.

So that shows that, in general, you can't just interchange limits. They don't commute.

 

[HallsofIvy]

I'm trying to understand how to interpret multidemensional limits. For example, suppose you have the following:

$$\lim\limits_{x \to \infty}\lim\limits_{y \to \infty} x\frac{1}{y}$$

Would this be infinity, 0, or 1?

None of the above! The limit simply doesn't exist.

This is really a more general version of the question I'm working with regarding the behavior of a function that has the property $$f_a(b) \to\limits_{a \to \infty} 1$$ and $$f_a(b) \to\limits_{b \to \infty} \infty$$ in the context that both a and b are going to infinity.

Thanks in advance for any help!

That means there are points arbitrarily far from the origin such that f is close to 1 and also points such that f is arbitrarily large. There is no one number (not even $\infty$) that the function gets close to. If there are two different limits by approaching a given point (even "the point at infinity" in two different ways, then the limit itself does not exist.