No misunderstanding. Unless you happen to disagree with 10000 years of mathematics. Let n be an arbitrary real number. Then, as you postulated: n/0 = ∞. Let m be another real number. Then 'obviously': m/0 = ∞. Wait... so, that obviously means n = m... all numbers are equal! Infinity is a bitch. Division by zero IS undefined on the field of real numbers, no easy way around it, period. Hell no, dear, not talking about one-sided limits, I'm not picky. The problem lies when we define a limit, as follows: Given f:R -> R and real-valued constants L and c, lim f(x) = L x -> c If, and only if, for every real ε > 0 there exists δ > 0 such that for every x where 0 < |x - c| < δ, it holds that |f(x) - L| < ε. Well, news flash for you: infinity is not a real-valued constant. Therefore it cannot be a limit. When we state that a limit 'is infinity', plus or minus, I don't care, we're making an informal assessment that the limit does not exist as the absolute value of the function increases without bound when approaching c.