# Limits in higher dimensions

1. Jul 28, 2010

### Quincy

1. The problem statement, all variables and given/known data
Do the following limits exist? State any relevant ideas.

a) limit as (x,y) -> (0,0) of (xy)/(x2 - y2)

b) limit as (x,y) -> (0,0) of (x2)/(3x4 + y2)

c) limit as (x,y) -> (0,0) of sin(2x)/y

3. The attempt at a solution

I don't really know where to start; I can't simplify them algebraically, how else can I determine the limit?

2. Jul 28, 2010

### Staff: Mentor

Does your book have any examples of limit problems like these? That would be a good place to start.

3. Jul 28, 2010

### Quincy

Yeah, but I'm having a hard time understanding the examples, that's why I came here for help.

4. Jul 28, 2010

### Staff: Mentor

Show us one of the examples and what you're having trouble understanding with it.

5. Jul 28, 2010

### whistler

Well, the limit exits if you approach the point from all possible paths and you get the same value. Its impossible do to try all combinations by hand but you can do some trial and error and find contradictions.

For example you let x=0, find lim y -> 0.
let y=0, find lim x -> 0.
let y=x, find lim x -> 0.
...

try diff paths until you find that one of the limits has a diff value from the rest, then you have a proof by contradiction, but if u have reason to believe the limit does exist, then its a bit different. Consult a calculus book.

6. Jul 28, 2010

### hunt_mat

For the first limit, try the two following limits for x and y, x_{n}=\sqrt{2}/n,y_{n}=1/n and see what sort of limit you get, then try x_{n}=\sqrt{3}/n,y_{n}=1/n. Are these two limits the same as n goes to infinity? If they're not the as whistler says, the limit doesn't exist.