Problem

For \(C(x)~=~20/(x~-~4)~+~5\), where \(x~\ge~0\) is measured in input-units and \(C\) in output-units, find the vertical asymptote and exclusion, state \(\text{both}~\text{one}\text{-}\text{sided}~\text{limits}\), and interpret the break in context.

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Algebra: denominator zero at x=4, but no vertical asymptote is recorded and x=4 is allowed. Limits: both one-sided limits are reported as finite 5. Context: input 4 is claimed feasible and the output is assigned the horizontal shift 5.

Algebra: denominator zero at x=4, which is treated as an attainable model value rather than an exclusion. Limits: x→4− gives C→−∞, and x→4+ gives C→+∞. Context: the unbounded outputs are claimed to be actual values at input 4.

Algebra: denominator zero and vertical asymptote at x=4; x≠4. Limits: x→4− gives C→+∞, and x→4+ gives C→+∞. Context: 4 lies on the nonnegative input scale but is excluded; unbounded outputs mark model breakdown.

Algebra: denominator zero and vertical asymptote at x=4; x≠4. Limits: x→4− gives C→−∞, and x→4+ gives C→+∞. Context: 4 lies on the nonnegative input scale but is excluded; unbounded outputs mark model breakdown.

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For \(C(x)~=~20/(x~-~4)~+~5\), where \(x~\ge~0\) is measured in input-units and \(C\) in output-units, find the vertical asymptote and exclusion, state \(\text{both}~\text{one}\text{-}\text{sided}~\text{limits}\), and interpret the break in context.

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