Final Answer:
The expression with the greatest value for all real values of x such that -1 < x < 0 is (1/x).
Step-by-step explanation:
In the given range -1 < x < 0, the expression (1/x) will have the greatest value. To understand this, let's analyze the behavior of the expression.
Consider the case where x is approaching 0 from the left side (as x gets closer to 0 but remains negative). As x becomes smaller, the denominator gets closer to zero, and the fraction (1/x) increases without bound, approaching negative infinity. Therefore, (1/x) grows larger and larger as x approaches 0 from the left.
On the other hand, if we examine the other expressions in the given options, they either approach zero or remain bounded. For instance, constants or expressions where x is in the denominator will decrease as x approaches 0 from the left. The reciprocal, (1/x), is the only expression that diverges towards negative infinity in this specific range, making it the one with the greatest value.
In mathematical terms, as x approaches 0 from the left side, lim (x → 0-) (1/x) = -∞, indicating an unbounded increase in the value of (1/x). This property establishes (1/x) as the expression with the greatest value in the given range.