Are the Limits in Higher Dimensions Solvable Algebraically?

[Quincy]

Homework Statement

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

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?

 

[Mark44]

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

 

[Quincy]

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

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

 

[Mark44]

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

 

[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.

 

[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.