1^inf is bad in the same way that 0*inf is bad. In fact, if you take the log of (1^inf), that's exactly what you get.
The catch is that although it is often convenient to manipulate 'inf' as if it were a number, it's not. An expression like '1/inf = 0' isn't really saying that inf is the reciprocal of 0; it's more shorthand for 'as x grows arbitrarily large, 1/x tends to 0'. Most instances where inf appears, we're really talking about limits or some such, and all the operations involved are really on *finite* numbers (though possibly an infinite set of all-finite numbers).
There are situations where it's consistent to say that 1inf = 1, just as sometimes it's consistent to say that 0*inf = 0. (limx=>inf0*x certainly equals 0.) But there are others where it breaks down; in the neighbourhood of x=1, y = inf, the function z=x^y is horrendously discontinuous.
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The catch is that although it is often convenient to manipulate 'inf' as if it were a number, it's not. An expression like '1/inf = 0' isn't really saying that inf is the reciprocal of 0; it's more shorthand for 'as x grows arbitrarily large, 1/x tends to 0'. Most instances where inf appears, we're really talking about limits or some such, and all the operations involved are really on *finite* numbers (though possibly an infinite set of all-finite numbers).
There are situations where it's consistent to say that 1inf = 1, just as sometimes it's consistent to say that 0*inf = 0. (limx=>inf0*x certainly equals 0.) But there are others where it breaks down; in the neighbourhood of x=1, y = inf, the function z=x^y is horrendously discontinuous.