Does the Limit of 3xy/((x^2)+(4y^2)) as (x, y) Approaches (0,0) Exist?

[manjum423]

Evaluate the limit or prove that it does not exist..

f(x,y) -> (0,0)
3xy/((x^2)+(4y^2))

The attempt at a solution:

Set x to 0 and you get 0
set y to 0 and you get 0
set y=x and you get 3x^2/5x^2 = 3/5
This means that limit does not exist.

Is this correct?
If this is correct, how do you know that you have to set y=x? Is there a generic approach to these kinds of problems?
Thank you for taking the time to read this and for your help.

 

[tiny-tim]

welcome to pf!

hi manjum423! welcome to pf!

(try using the X² button just above the Reply box )

Set x to 0 and you get 0
set y to 0 and you get 0
set y=x and you get 3x^2/5x^2 = 3/5
This means that limit does not exist.

Is this correct?

completely

If this is correct, how do you know that you have to set y=x? Is there a generic approach to these kinds of problems?

you try y = kx first (for constant k),

if that doesn't work, try y = kxⁿ

if that doesn't work, assume the limit exists, and try to prove it!

alternatively, use polar-ish coordinates, eg x = 2rcosθ, y = rsinθ, giving … ? ​

 

[SammyS]

Evaluate the limit or prove that it does not exist..

f(x,y) -> (0,0)
3xy/((x^2)+(4y^2))

The attempt at a solution:

Set x to 0 and you get 0
set y to 0 and you get 0
set y=x and you get 3x^2/5x^2 = 3/5
This means that limit does not exist.

Is this correct?
If this is correct, how do you know that you have to set y=x? Is there a generic approach to these kinds of problems?
Thank you for taking the time to read this and for your help.

Hello manjum423 . Welcome to PF !

We do like you to use the supplied homework template.

Homework Statement

Homework Equations

The Attempt at a Solution

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You seem to have some difficulty writing you problem. It appears that you problem is something like:

Evaluate the limit or prove that it does not exist..

Lim_{(x,y)→(0,0)} 3xy/((x²)+(4y²))
​

Which can be displayed more nicely using LaTeX.

$\displaystyle \lim_{(x,y)\to(0,0)} \frac{3xy}{(x^2)+(4y^2)}$

Your solution is correct.

For your problem, the method you used works fine.

You could make it a bit more general by approaching the origin along the arbitrary line , y = mx .

That method doesn't always work either.

One pretty good scheme is to change to polar coordinates. Then only one variable, r, goes to zero. The other, θ, remains arbitrary.