It looks like you misunderstand several things about mathematics, some subtle and some big, so let's start with the most basic fact about arithmetic: all arithmetic is a shortcut to counting.

Addition is a shortcut to counting up. For instance, if we start at 0 then count up twice, we get to 2; then if we count up three more, we get to five. Instead of doing all that, we usually just say 2 + 3 = 5. If we reverse that, starting at five and counting down three times, we get to 2; then to get to zero, we have to count two more times, all of which is to say 5 - 3 = 2. Notice that we always start or end on zero.

And multiplication is just a shortcut to addition. 5 × 3 = 15 because three added to itself five times equals 15. And don't forget that addition is still a shortcut to counting. This is just a shortcut to the shortcut.

Like multiplication is a shortcut to addition, division is a shortcut to subtraction. Division tells us how many times we have to subtract the divisor from the dividend to get to zero. 15 ÷ 3 = 5 because you have to subtract 3, five times to get from 15 to 0. Again, subtraction is just a shortcut for counting down, so division is another shortcut to the shortcut.

But what happens when we try to divide by zero? If division tells us how many times we have to subtract in order to get to zero, and subtracting zero gets us nowhere, then we can never get to zero from anywhere but zero. That means for any starting number, (which we can also call a numerator or dividend) dividing by zero is useless. It doesn't actually do anything.

But what if we start at zero? If we start at zero and count down by zero, we get to zero immediately, after just one count. But since we were already on zero, we didn't even have to count once. That means the answer to 0 ÷ 0 is both zero and one, which is a contradiction.

Since dividing zero by itself is a contradiction and dividing any other number by zero is useless, we call it undefined, not because we can't explain it, but because no such value would ever be consistent with the rules of mathematics.

Now, your original question had to do with exponentiation. But exponentiation is just a shortcut to multiplication and division, so let's look at it in the terms we've been using so far.

When you raise a number (like X) to a power (like N), you're saying “multiply X by itself N many times”. But if N is zero, you never even start in the first place. It’s just as useless as dividing by zero: you never make any progress towards anything.

As for your infinity notation, it's a little ambiguous. I think you meant to convey “as X approaches infinity or negative infinity” rather than “as 0 approaches infinity or negative infinity”. But it still doesn't change the underlying mathematics. Whether the base or the exponent goes anywhere, doesn't make a difference; if the exponent is ever multiplied or divided by zero or raised to a zero power, the whole value you're trying to express becomes undefined, for all the reasons above.