However, any function where the value x approaches a certain limit and produces a function value 0/0, can be changed or derived to a further to produce a non-zero answer.
So 0/0 can have some application.
ex.
lim (x --> 2) ((x^3)-8)/(x-2) = 0/0
((x^3)-8)/(x-2) = (x-2)(x^2+2x+4)/(x-2) = (x^2)+2x+4
lim (x--> 2) (x^2)+2x+4 = 2^2 + 2(2) + 4 = 12.
Rule #1 of math: "You shall not divide by zero!"
So if you try to divide anything with zero, even zero itself, you violate the rule #1. If you do it anyway you can get any answer, i.e. you cannot ever trust the answer.
We have a function f(x) = x/x. What is f(0)?
f(0) = 0/0 = 1 because x/x is always = 1. Right?
We have another function f(x) = 0/x. What is f(0)?
f(0) = 0/0 = 0 because 0/x is always = 0. Right?
So now I have prooved that 0/0 is both =1 and =0. Right?
Wrong. I divided by zero in both cases, thus violated rule #1, and got a unreliable result. The proofs are worthless.
Conclusion: Do not ever try to divide by zero!