0^0

[Albert Chan]

(11-24-2023 09:41 PM)Albert Chan Wrote:  Limit depends on relative rate of base and exponent shrink to 0
For example, we have identity: ε ^ (1/ln(ε)) = e

If absolute of exponent shrink slower, we get 0^0=0
Cas> limit(ε ^ (1/(-ln(ε))^0.99999), ε, 0, 1)      → 0

If absolute of exponent shrink faster, we get 0^0=1
Cas> limit(ε ^ (1/(-ln(ε))^1.00001), ε, 0, 1)      → 1

Cas> assume(p > 0)
Cas> limit(ε ^ (ε^p), ε, 0, 1)      → 1

ε^0, ε^√ε, ε^(ε*(polynomial of ε)) ... all have 0^0=1
This explained why define 0^0=1 is so useful, even though it is not the truth.