Definitions for indeterminate forms

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In this thread I will define values for the 7 indeterminate forms, starting with giving values to the most intuitive forms and using them to build up a logically consistent system that matches with our intuition.

1^? = 1
0^0 = 1
?^0 = 1
0*? = 0
?/? = 0
0/0 = 0
?-? = 0

1^?
multiplying by 1 is the same as dividing by 1, so 1^? = 1/(1^?), the only number that is equal to it's multiplicative inverse is 1, so we have
1^? = 1

0^0
30 = 2*3*5, how many times did we multiply by 0? 0 times. if we have 0^0 = x then we have 30 = 30x, which only makes sense if x=1
0^0 = 1

1/?
1/x gets arbitraily close to 0 as |x| gets larger
if a>b, 1/a is closer to 0 than 1/b
? is bigger than any real number, so 1/? must be closer to 0 than any arbitrarily small distance, so it must be 0
1/? = 0

?*0
if 0^0 = 1 then
1^? = 0^(?*0)
0*? = 0 implies 1^? = 1
0*? > 0 implies 1^? = 0
0*? < 0 implies 1^? = ?
we know 1^? = 1, so ?*0 = 0
?*0 = 0

?^0
same argument as 0^0, 2*3*5 = 30, we multiplied by ? 0 times, which is the same as multiplying by 1.
?^0 = 1

?/?
?/? = ? * 1/? = ?*0 = 0
?/? = 0

0/0
0/0 * 0 = (0*0)/0 = 0/0
multiplying by 0 didn't change the value, and 0 multiplied by anything is 0, so 0/0 must be 0
0/0 = 0

?-?
?-? = ?(1 - 1) = ?*0 = 0
?-? = 0