The limit of the division function under certain conditions is not the same as division - division has a discontinuity at 0 which is expressing the same thing.
Defining division by zero only works if you don’t care to preserve the field axioms, which is often inconvenient and so not done. The Riemann sphere is not a field, and fairly niche in the context of mere division, so I stand by my accusation of this being misleading.
The policy of not defining division by zero to preserve the cancellation law is the most sensible default.
We know the limit of a/x as x --> 0 if a ≠ 0.
The limit is different if a = 0.
The limit of the division function under certain conditions is not the same as division - division has a discontinuity at 0 which is expressing the same thing.
Defining division by zero only works if you don’t care to preserve the field axioms, which is often inconvenient and so not done. The Riemann sphere is not a field, and fairly niche in the context of mere division, so I stand by my accusation of this being misleading.
The policy of not defining division by zero to preserve the cancellation law is the most sensible default.