Proving a limit for a multi-variable equaton

[bosox09]

Homework Statement

Using the definition of a limit, prove that

lim(x, y) --> (0,0) (x^2*y^2) / (x^2 + 2y^2) = 0

Homework Equations

Now, I know that the limit of f(x, y) as (x, y) approaches (a, b) is L such that lim (x, y) --> (a, b) f(x, y) = L. Also, for every number epsilon > 0, there is a delta > 0 such that |f(x, y) - L| < epsilon.

I believe the above is the definition of a limit of two variables.

The Attempt at a Solution

In a sense, f(x, y) --> L (two VALUES) as (x, y) --> (a, b) (two POINTS). By making the distance between points (x, y) and (a, b) extremely small (some value epsilon), we make the distance between f(x, y) and L (some value delta) subsequently small. For any interval [L - epsilon, L + epsilon], there is a subsequent plane with center (a, b) and radius delta > 0 satisfying this.

What I want to do is use some very small value of epsilon to find a value of delta that satisfies the definition of a limit.

 

[HallsofIvy]

Generally, for problems like this, the best thing to do is to change to polar coordinates: $x= rcos(\theta)$, $y= rsin(\theta)$ because then r alone measures how close to (0,0) we are. If the limit as r goes to 0 is independent of $\theta$, then the limit exists and is that value. If the limit as r goes to 0 depends on $\theta$ then the limit of the function does not exist.

 

[sutupidmath]

$$|\frac{x^2y^2}{x^2+2y^2}-0|=|\frac{x^2y^2}{x^2+2y^2}|=|\frac{x^2}{x^2+2y^2}y^2|=|\frac{x^2}{x^2+2y^2}|y^2<y^2<x^2+y^2<\delta^2=\epsilon$$

So for a choice of :

$$\delta=\sqrt{\epsilon}$$ it would work out.


$$x^2<x^2+2y^2=>\frac{1}{x^2+2y^2}<\frac{1}{x^2}=>\frac{x^2}{x^2+2y^2}<1$$


$$x^2+y^2<\delta^2$$

edit: Halls suggestion is correct and helpful no doubt. but you have more options now!

 

[bosox09]

Thank you, you guys were very helpful!