Why is 0⁰ = 1?

[LinuxGal]

Because that's the trend.

[polymath257]

More generally, if f(x) and g(x) are analytic functions with f(0)=g(0)=0, then the limit of f(x)^g(x) as x-->0 is 1. Your calculation was with f(x)=g(x)=x.

But if you go away from analytic functions, that may no longer be the case. So, if f(x)=e^(-1/x^2) and g(x)=x^2, then
as x-->0, both f(x) and g(x)-->0, but f(x)^g(x) =e^(-1) is not 1.

In many situations, it is convenient to have the *convention* that 0^0 =1 to make certain formulas correct when one variable is 0.

But, for limits, you should always regard 0^0 as indeterminate.

[Jehanne]

0 to the power of 0 is undefined.

[LinuxGal]

More generally, if f(x) and g(x) are analytic functions with f(0)=g(0)=0, then the limit of f(x)^g(x) as x-->0 is 1. Your calculation was with f(x)=g(x)=x.

But if you go away from analytic functions, that may no longer be the case. So, if f(x)=e^(-1/x^2) and g(x)=x^2, then
as x-->0, both f(x) and g(x)-->0, but f(x)^g(x) =e^(-1) is not 1.

In many situations, it is convenient to have the *convention* that 0^0 =1 to make certain formulas correct when one variable is 0.

But, for limits, you should always regard 0^0 as indeterminate.

If you take the limit of 0^x from the right, you get 0. From the left, you get infinity. If you take the limit of x^0 you get 1. So it's a case of choose your own adventure.

[polymath257]

More generally, if f(x) and g(x) are analytic functions with f(0)=g(0)=0, then the limit of f(x)^g(x) as x-->0 is 1. Your calculation was with f(x)=g(x)=x.

But if you go away from analytic functions, that may no longer be the case. So, if f(x)=e^(-1/x^2) and g(x)=x^2, then
as x-->0, both f(x) and g(x)-->0, but f(x)^g(x) =e^(-1) is not 1.

In many situations, it is convenient to have the *convention* that 0^0 =1 to make certain formulas correct when one variable is 0.

But, for limits, you should always regard 0^0 as indeterminate.

If you take the limit of 0^x from the right, you get 0. From the left, you get infinity. If you take the limit of x^0 you get 1. So it's a case of choose your own adventure.

Which is why it is said to be indeterminate.

[UniversesBoss]

It's zero.