Answer: Limit of (1 - (x^.5))/(1 + (x^.5)) as x Approaches Neg. Infinity

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The limit of the expression (1 - (x^.5))/(1 + (x^.5)) as x approaches negative infinity is definitively -1 when considered in the context of complex numbers. The discussion highlights that the square root of a negative number is imaginary, which is crucial for understanding the behavior of the function as x moves towards negative infinity. Val's initial confusion stemmed from attempting to evaluate the limit within the confines of real numbers, where the square root of negative values is undefined. The conclusion emphasizes the importance of recognizing the transition from real to complex analysis in limit problems.

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So, I've been mulling over this limit problem for far too long. I feel completely at a loss and refuse to accept my answer of "no limit" or "undefined" regarding the following:

The limit of (1- (x^.5))/(1+(x^.5)) as x approaches NEGATIVE infinity.

Someone care to elaborate on what is actually going on here? As I see it, even multiplying by the conjugate, I still end up with an x variable under a radical and so, the square root of any negative number is imaginary. Can such a number "approaching negative infinity" or "x" can also be a positive number, but moving in the negative direction? Being that as it may, I'm trying to keep the answer within the context of real numbers. So, what is the answer?

-Val-
 
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You can't keep x^(0.5) for x negative in the context of the real numbers. Face it. Once you gotten over that, it certainly has a limit in the complex numbers, regardless of branch choice. The answer is -1.
 
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