My question is about Calculating of limitation of (X/Y)?

Question

lim(x--->0){lim(y---->0){ x / y } }
x & y is independent variable

Answer 1

In this case, you would just calculate the limits from the inside out. That is, first calculate [y→0]lim (x/y) for some fixed x, and then take the result from that calculation and take the limit of it as x→0. Note that what answer you will get (or rather, whether you will get an answer) depends somewhat on the topology you are using, and in particular, which points at infinity it contains. On R (which has no points at infinity), this limit does not exist -- ∀x≠0, [y→0]lim (x/y) does not exist, so the domain of [y→0]lim (x/y) regarded as a function of x is just {0}. Since this domain has no accumulation points, it is meaningless to ask what the limit of this function is as x→0, or any other point.

If working in the extended real line, this limit still does not exist, since if x>0, we have [y→0⁺]lim (x/y) = ∞ and [y→0⁻]lim (x/y) = -∞, so [y→0]lim (x/y) = ∞ doesn't exist, and if x<0, we have [y→0⁺]lim (x/y) = -∞ and [y→0⁻]lim (x/y) = ∞, so [y→0]lim (x/y) still doesn't exist. Thus the domain of [y→0]lim (x/y) regarded as a function of x is still only {0}, and it still has no limit as x→0 or any other point.

On the other hand, if working in the real projective line or the Riemann sphere (where there is only one ∞, and there is no distinction between ∞ and -∞), then the story is entirely different. Here, we have that ∀x≠0, [y→0]lim (x/y) = ∞, so this function is defined everywhere, and is equal to infinity everywhere except 0 (at x=0, of course, [y→0]lim (x/y) = [y→0]lim 0 = 0, but this is not pertinent to the limit of [y→0]lim (x/y) as x approaches 0). Therefore [x→0]lim ([y→0]lim (x/y)) = ∞.

Note that in this case the order of the limits is extremely important -- in any of the four structures mentioned above, [y→0]lim ([x→0]lim (x/y)) = [y→0]lim 0 = 0 ≠ ∞. Also note that both of these are different from calculating [x→0, y→0]lim (x/y), since in that case x and y would approach 0 at the same time, rather than having y approach 0, and then x, or vice versa. In fact, [x→0, y→0]lim (x/y) never exists, since x/y can be made to approach many different values, depending on which path x and y approach 0 along.

Answer 2

Check L'Hopital Rule. It deals with with the indeterminate form 0 over 0.

Answer 3

The limit of x/y as (x,y) -> (0,0) does not exist. Suppose (x,y) -> (0,0) along the line y = x; then the limit is 1.

But if (x,y) -> (0,0) along x = 0, then the limit is 0

If a limit exists, it is independent of the path to the limiting argument. Therefore, this limit DNE.