Problem

For \(A(x)~=~8~+~500/x\), where \(x\) is a positive whole-number production amount and \(A\) is average cost in dollars per item, state the large-input limit, the mathematical right-hand limit near \(\text{zero}\), and the contextual interpretation for the discrete domain.

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Large input: as x→∞, A→8 dollars/item. Near zero: A→8 mathematically as x→0+. Context: zero is feasible; long-run average cost is exactly 8 dollars/item.

Large input: as x→∞, A→0 dollars/item. Near zero: A→∞ mathematically as x→0+. Context: zero is not discretely sampled; long-run average cost approaches 0 dollars/item.

Large input: as x→∞, A→8 dollars/item. Near zero: A→∞ mathematically as x→0+. Context: zero is not discretely sampled; long-run average cost approaches 8 dollars/item.

Large input: as x→∞, A→∞ dollars/item. Near zero: not reported. Context: production is treated as continuous; no finite long-run average cost is identified.

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For \(A(x)~=~8~+~500/x\), where \(x\) is a positive whole-number production amount and \(A\) is average cost in dollars per item, state the large-input limit, the mathematical right-hand limit near \(\text{zero}\), and the contextual interpretation for the discrete domain.

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