![]() ![]() We know they will never output anything greater than 1, or less than -1, we are even able to compute them for any real number. One of the main things a function has to do to approach a number is to start to stabilize. Not all functions approach a number as their input approaches infinity. So we have an indeterminate form when we have a base approaching 1 and exponent approaching infinity, but not when we have a base that EQUALS 1 and exponent approaching infinity. We no longer have an infinitesimal increment away from 1 that can be overpowered by the increase of the exponent. The expression 1^b is always 1, no matter how large or small the exponent. HOWEVER, if a is not some function that approaches 1, but is actually the number 1, then we no longer have an indeterminate form. So that is why we say 1^∞ is an indeterminate form. ![]() If we find that a approaches 1 and b approaches infinity, we have an indeterminate form, because we can't tell without further analysis whether the forces attracting a toward 1 (making the expression approach 1) are overpowered by the forces moving b toward infinity (making the expression approach infinity or zero, depending on whether a is slightly greater than or less than 1).
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