Find Limit at Origin for ((x)(y^2))/((x^2)+(y^4))

[jcook735]

find the limit if it exists or show that the limit does not exist for:

limit(x,y) ---> ((x)(y^2))/((x^2)+(y^4)) as (x,y) ----> (0,0)


I took the limit approaching the origin along the x-axis and y axis.
So, took the limit as (x,y) ---> (x,0) and came up with 0.
Then, I took the limit as (x,y) ---> (0,y) and came up with 0 again.

Is 0 my limit or am I missing something?

 

[╔(σ_σ)╝]

No, you are not missing anything. Your answer is correct.

 

[jcook735]

WOOOO thanks

 

[HallsofIvy]

No, unfortunately you are missing a lot! Once you move beyond one dimension, the limit as you approach the "target" point, from any direction must be the same. Most Calculus texts have examples of that in 2 dimensions but the same is true for 3 dimensions. Just getting the same result approaching (0, 0, 0) along the coordinate axes, or even along every straight line, does NOT imply you will get the same thing along any curve.

(Hold on, I just realized that, although you titled this "Finding limits is R³", it is really a problem in 2 dimensions, not 3. Oh, well, same thing is true.)

If, for example, you take the limit as (x, y) goes to (0, 0) along the parabola x=y^2, the limit becomes
\lim_{y\to 0}\frac{(y^2)(y^2)}{(y^2)^2+ y^4)}= \lim_{y\to 0}\frac{y^4}{2y^4}= \lim_{y\to 0}\frac{1}{2}= \frac{1}{2}.

Since that is different from the limit as you approach (0, 0) along the axes (or, in fact, along any straight line), the limit does not exist!

 

[jcook735]

thanks! I talked to my teacher about this and she agrees with you HallsOfIvy. I see it now