How to find limit of multivariable function?

[Puchinita5]

Homework Statement

Hi. I'm going to provide two homework examples. The first one I got correctly. But the second one, I used the same method as the first question but I got the answer incorrect. I'm really confused.

The first question was
$$\lim_{(x,y)\to(0,0)}\frac{(xycosy)}{3x^2+y^2}$$
f(x,0)=0
f(0,y)=0
for y=x , f(x,x)= (1/4)cosx

since the three limits are not equal, the limit does not exist!

Okay now the second question:

$$\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}$$

I again did:
f(x,0)=0
f(0,y)=0
$$f(x,x)=\frac{x^2}{x\sqrt{2}}$$

since they aren't the same, limit does not exist. I THOUGHT.
I looked in the solutions manual, and it says "We can see that the limit along any line through (0,0) is 0, as well as along other paths through (0,0) such as x=y^2. So we suspect that the limit exists and equals 0; we use the Squeeze Theorem to prove our assertion.

$$0\leq\left | (\frac{xy}{\sqrt{x^2+y^2}})\right |\leq \left |x \right | since \left |y\right |\leq \sqrt(x^2+y^2) and \left |x\right | \to \0 \ as (x,y) \to \(0,0).\ so\ the\ limit\ equals\ 0.$$

I guess I don't understand how you choose the method to solve for the limit. Is there something about the function that tells you what to do? Why is f(x,x) not an option?

I mean, I guess it makes sense because I'm sure you can't just plug in anything or at some point you will get something that won't equal the same thing. But how do you know what to do? How would I have known to use the squeeze theorem?

I'm so confused.
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[Puchinita5]

this always happens to me...the SECOND i post something on this site i realize something.

I am supposed to plug in a zero into the (x^2)/(x2^.5) aren't I?
which would also make it equal to 0.

In which case...can someone just explain to me the logic behind the squeeze theorem in this example? It still confuses me.

And if you test out different functions, such as f(x,x) or f(x,0) etc... and get the same limit for all of them, do you always need to use the squeeze theorem to verify that it is indeed the limit?

 

[LCKurtz]

Limits of functions of two variables can be very tricky. The problem is that there is not a standard approach that you can count on work any problem. So, given a problem where you don't know whether the limit exists, the usual thing is to check to see if you can get different answers along different paths. If you can, that is usually pretty easy and it settles the problem giving that the limit doesn't exist.

If you keep getting the same answer along many different paths, that may be because the limit does exist and is equal to that common number. Proving it is then where the difficulty usually arises. If you suspect, for example, that the limit is 0, then you can try to overestimate the function with something easier that does go to 0. That is the idea of the squeeze principle. You will learn a few techniques along the way to try on such problems after which they won't seem quite so intimidating. Don't get too hung up on it if you are having trouble with these at first. Experience and seeing more examples will help. And your course will soon move on to other more enjoyable topics.