Originally posted by econundrum
First of all, thank you for the first fact you pointed out, that infinity^0 is not 1. I had heard that before, but I had never seen a situation where infinity^0 was not 1, so I had assumed it to be wrong. (My teacher didn't even catch ...[text shortened]... who replied with useful information! It definately helped a lot.
the second part has been addressed in a previous post so i'll not go into that now.
you might think of "x
raised to the zero power" as "x
". it's unity for all x except x
= 0 and x
= inf, where it is undefined because the quotient can be anything. (again, "zero times what equals zero?" and "infinity times what equals infinity?" ).
here are three examples to show why inf/inf cannot be assigned a specific value.
1. lim x
) = + infinity becaue it simplifies to (3/2)^x
2. the reciprocal of the above, lim x
= 0, as (2/3)^inf is zero.
) = exp(2) (e
since we have found three different values for inf/inf, that quantity cannot be assigned a definite value.
for inf^0, one example that comes to mind is (exp(x
)), which goes to exp(1).
hope this isn't too confusing.