The other comments are correct: 
is undefined. Similarly, the limit of 
as 
approaches 
is also undefined. However, if you take the limit of 
as 
approaches zero from the left or from the right, you get negative and positive infinity respectively.
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The other comments are correct: is undefined. Similarly, the limit of as approaches is also undefined. However, if you take the limit of as approaches zero from the left or from the right, you get negative and positive infinity respectively.
does tend to 
as you approach zero from the left, and 
as you approach from the right:
That these limits are not equal is why 
is undefined.
Like 10/2- imagine a 10 square foot box, saying 10 divided by 2 is like saying “how many 2 square foot boxes fit in this 10 square foot box?” So the answer is 5.
But if you take the same box and ask “how many boxes that are infinitely small, or zero feet squared, can fit in the same box the answer would be infinity not “undefined”. So 10/0=infinity.
I understand why 2/0 can’t be 0 not only because that doesn’t make and since but also because it could cause terrible contradictions like 1=2 and such.
Ah math is so cool. I love infinity so if anyone wants to talk about it drop a comment.
Edit: thanks everyone so much for the answers. Keep leaving comments though because I’m really enjoying seeing it explained in different ways. Also it doesn’t seem like anyone else has ever been confused by this judging by the comment but if anyone is I really liked this video https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:foundation-algebra/x2f8bb11595b61c86:division-zero/v/why-dividing-by-zero-is-undefined
I think it does because you can derive it from infinity/0 times 0/infinity.
It is worth noting that since $\infty$ is not a number, the expression $\frac{1}{\infty}$ is not meaningful. That is, you cannot evaluate this expression. It is not a number. You can think of it in much the same way as $\frac{1}{0}$ --- an expression without any interpretation.
The only means that we have to talk about expressions involving $\infty$ is through the concept of limits. The shorthand $$ \lim_{x \to \infty} \frac{1}{x} = \frac{1}{\infty} = 0 $$ is sometimes used in a careless way by students, but the middle expression in this string of equations isn't strictly meaningful. It is only a device used to remind you that we are looking at what happens as we divide 1 by increasingly large numbers.
So, I suppose that in some sense, you can say that $\frac{1}{\infty} \neq 0$, as the left-hand side here isn't any number at all.
Your argument makes use of stochastics. If you want to do things the mathematically exact way, you would first need to define a probability measure on $(0,\infty)$. Assuming that you choose a continuous distribution, then yes, the Probability that you hit $5$ is $0$ - but there are infinitely many numbers, so there is no contradiction. (Keep in mind that $0 \cdot \infty$ is not well defined, and unless in the case of $\frac1{\infty}$ there is no heuristical way to define it)
Check out Measure Theory: https://en.wikipedia.org/wiki/Measure_%28mathematics%29
sorry if the question doesnt make sense i havent been invested in math theory for long as ive only taken alg 2 and minor precalc but why is it that one over infinity equals zero rather than an infinitely small finite number? from my thoughts i feel as if it cant be zero because if you have anumerator there is a value no matter the size of a denominator, almost like an asymptotic relationship with the value reaching closer to zero but never hitting it. i understand zero is a concept so you cant operate with it so you cant exactly create a proof algebraicly but then how could you know it equals zero? just need second thoughts as its a comment debate between me and my brother. many thanks!
edit: my bad i wasnt very misunderstood on alot of things and the question was pretty dumb in hindsight, my apologies