My math teacher wants me to submit the answer in the class page by 9:30!
Not sure what math class you are in, so the answer may be inappropriate, but here goes:
(infinity)^zero is really undefined
But the way this expression comes up in practice is always the limit as something goes TOWARD
infinity or TOWARD zero, or both. For example, take x to the 0 power, and let x get bigger and bigger.
Then the answer each time is 1, so the limit is also 1.
But take infinity to the y power, and let y get smaller and smaller.
Then the answer is infinity each time, so the limit is also infinity.
The interesting case is where x and y are related in some way, so x
gets bigger as y gets smaller in a coordinated way. Depending on the
algebraic relation between x and y, the answer can come out 1 or
infinity or anything in between.
infinity or TOWARD zero, or both. For example, take x to the 0 power, and let x get bigger and bigger.
Then the answer each time is 1, so the limit is also 1.
But take infinity to the y power, and let y get smaller and smaller.
Then the answer is infinity each time, so the limit is also infinity.
The interesting case is where x and y are related in some way, so x
gets bigger as y gets smaller in a coordinated way. Depending on the
algebraic relation between x and y, the answer can come out 1 or
infinity or anything in between.
That's why it is really undefined.
Hope that make sense