Problem

For \(D(t)~=~t/(t~-~10)\), where \(t~\ge~0\) is elapsed time in seconds and \(D\) is unitless, 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 and vertical asymptote at t=10; t≠10. Limits: t→10− gives D→−∞, and t→10+ gives D→+∞. Context: 10 seconds is on the timeline but excluded; the asymptote marks model breakdown between two branches.

Algebra: denominator zero at t=10, but no vertical asymptote is recorded and t=10 is allowed. Limits: both one-sided limits are reported as finite 1. Context: 10 seconds is claimed feasible and assigned the long-run ratio 1.

Algebra: denominator zero at t=10, which is treated as an attainable model value rather than an exclusion. Limits: t→10− gives D→−∞, and t→10+ gives D→+∞. Context: the unbounded outputs are claimed to be actual values at 10 seconds.

Algebra: denominator zero and vertical asymptote at t=10; t≠10. Limits: t→10− gives D→+∞, and t→10+ gives D→+∞. Context: 10 seconds is on the timeline but excluded; the asymptote marks model breakdown between two branches.

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For \(D(t)~=~t/(t~-~10)\), where \(t~\ge~0\) is elapsed time in seconds and \(D\) is unitless, 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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