I am being taught that 1^(infinity) is indeterminate. But I couldn't stop thinking why 1x1x1x1....=1 is incorrect.
Please tell me if my reasoning is correct?
I thought like this. So, for limit to exist, LHL must be equal to RHL. left hand limit and right hand limit came different for 1^(infinity). For LHL, I took a value slightly lower than 1, which is now a fraction. And (Fraction)^(infinity) is 0 (zero). For RHL, I took number slightly greater than 1, which is NOT a fraction, and hence its power raised to infinity is infinity. So LHL and RHL are not equal.
Hence, indeterminacy arises only at 1 and not at numbers like 2, 3 etc. Because, then LHL and RHL will both be equal.
Is my Reasoning Correct? Is there a theoretical way to prove it?