Can you actually prove that ? Because (and I might be wrong, its been A WHILE since I did math), I recall a theorem saying something along the lines of :
if the lim[x->0, x<0] (1/x) = lim[x->0, x>0] (1/x) then you have 1/0 = said limit.
There were probably some extra rules. Probably something to do with continuity or something, I don't recall
But that simple rule isn't even fulfilled, since
lim[x->0, x<0] (1/x) = -infinity
lim[x->0, x>0] (1/x) = +infinity
So i'm pretty much sure you can't prove that, and more importantly, its mathematically incorrect.
I believe that you are confusing it with one of the indeterminate forms. 1/infinity is not indeterminate, the limit as x approaches infinity from either the right or the left yields the same result (I think that is what you were going for). For example, take the multiplicative inverse of some arbitrarily large number and then an arbitrarily larger number. You will notice that the value you get by choosing a larger number will always be smaller. Therefore, you can demonstrate that 1 (or any number you choose, for that matter) divided by infinity is zero, and can even actually define zero as such.
Yeah, I'm not disagreeing with the 1/infinity = 0, that part is mathematically true, and can be proven the way you did. I'm disagreeing with the part you are supposed to prove, which is 1/0 = infinity, since its either +infinity or -infinity depending on which side you approach 0
Do essentially the opposite. 1/ arbitrarily small numbers grows larger the smaller the denominator. As x approaches 0, the limit approaches infinity, from either the right or the left. Therefore, you can demonstrate the other statement is also as true as any math statement can be (it is also not an indeterminate form).
Except its not.
Depending on which side of 0 you approach, 1/arbitraly small numbers grows larger OR smaller indefinitely.
So the limit on 0+ is +infinity, and the limit on 0- is -infinity. So you can't say 1/0 = infinity
I do have a theory that dividing anything by zero equals infinity:
1. The larger the number, the closer its reciprocal is to zero, which means the reciprocal of zero itself is larger than anything, hence infinity.
2. The horizontal line on a graph has a slope of zero, vertical line is perpendicular to horizontal line, which means its slope is the negative reciprocal of zero, infinity.
3. It's impossible to get any number other than zero by multiplying a number by zero, which means that number has to be infinitely big (or small)
4. No matter how many zeros you add together, the sum can't be a number other than zero, which means you need to add an infinite amount of zeros together.
5. For Example, using the equation v=d/t, If something moved a certain distance in a time of zero seconds it means it moved without time change, which means its speed is bigger than any number, which means infinity.
I can't explain things that clearly, hopefully someone understands what I'm saying.
if the lim[x->0, x<0] (1/x) = lim[x->0, x>0] (1/x) then you have 1/0 = said limit.
There were probably some extra rules. Probably something to do with continuity or something, I don't recall
But that simple rule isn't even fulfilled, since
lim[x->0, x<0] (1/x) = -infinity
lim[x->0, x>0] (1/x) = +infinity
So i'm pretty much sure you can't prove that, and more importantly, its mathematically incorrect.
Depending on which side of 0 you approach, 1/arbitraly small numbers grows larger OR smaller indefinitely.
So the limit on 0+ is +infinity, and the limit on 0- is -infinity. So you can't say 1/0 = infinity
1. The larger the number, the closer its reciprocal is to zero, which means the reciprocal of zero itself is larger than anything, hence infinity.
2. The horizontal line on a graph has a slope of zero, vertical line is perpendicular to horizontal line, which means its slope is the negative reciprocal of zero, infinity.
3. It's impossible to get any number other than zero by multiplying a number by zero, which means that number has to be infinitely big (or small)
4. No matter how many zeros you add together, the sum can't be a number other than zero, which means you need to add an infinite amount of zeros together.
5. For Example, using the equation v=d/t, If something moved a certain distance in a time of zero seconds it means it moved without time change, which means its speed is bigger than any number, which means infinity.
I can't explain things that clearly, hopefully someone understands what I'm saying.