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

• jcook735
In summary, the limit (x,y) ---> ((x)(y^2))/((x^2)+(y^4)) as (x,y) ----> (0,0) does not exist, as the limit approaching the origin along different directions gives inconsistent results. Approaching along the axes gives a limit of 0, while approaching along the parabola x=y^2 gives a limit of 1/2, showing that the limit depends on the path taken to approach the origin.
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?

WOOOO thanks

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!

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

## 1. What is the limit at the origin for the function ((x)(y^2))/((x^2)+(y^4))?

The limit at the origin for this function is 0. This can be determined by plugging in 0 for both x and y, which results in 0/0. By using algebraic manipulation and factoring, the limit can be simplified to 0.

## 2. How do you approach finding the limit at the origin for this function?

To find the limit at the origin, one can use the method of substitution by plugging in 0 for both x and y and simplifying the resulting expression. Another approach is to use algebraic manipulation and factoring to simplify the function and then plug in 0 for both x and y.

## 3. Can the limit at the origin for this function be solved using L'Hopital's rule?

No, L'Hopital's rule cannot be used to find the limit at the origin for this function. This rule can only be applied when the limit is in an indeterminate form, such as 0/0 or infinity/infinity, and this is not the case for this function at the origin.

## 4. Is the limit at the origin for this function the same as the limit at other points?

No, the limit at the origin for this function may be different from the limit at other points. The limit at the origin is only a specific case, and the limit at other points may require a different approach to solve.

## 5. What is the significance of finding the limit at the origin for this function?

Finding the limit at the origin for this function can help in understanding the behavior of the function near the origin. It can also be used to determine if the function is continuous at the origin or if there is a point of discontinuity. Additionally, it is an important concept in calculus and is used in various applications such as optimization and curve sketching.

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