A patient receives a 150 -mg injection of a drug every 4...

A patient receives a 150 -mg injection of a drug every 4 hours. The graph shows the amount $f(t)$ of the drug in the bloodstream after t hours. Find $$\lim _{t \rightarrow 12^{-}} f(t) \quad \text { and } \quad \lim _{t \rightarrow 12^{+}} f(t)$$ and explain the significance of these one-sided limits.

A patient receives a 150 -mg injection of a drug every 4 hours. The graph shows the amount $f(t)$ of the drug in the bloodstream after t hours. Find
$$\lim _{t \rightarrow 12^{-}} f(t) \quad \text { and } \quad \lim _{t \rightarrow 12^{+}} f(t)$$
and explain the significance of these one-sided limits.

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Key Concepts

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only – from the left (denoted as a limit from below) or from the right (denoted as a limit from above). They are essential for understanding the behavior of functions at points where there may be a discontinuity or sudden change.

Jump Discontinuity

A jump discontinuity occurs when the left-hand limit and the right-hand limit of a function at a certain point exist but are not equal. This indicates a sudden change in the function’s value, which is especially important in models where abrupt interventions or changes occur.

Pharmacokinetic Modeling in Intermittent Drug Dosing

In the context of drug dosing, the model often involves periodic injections where the drug's concentration in the bloodstream decays over time between doses. The instant when a new dose is given can create a jump discontinuity in the graph of the drug amount, with the one-sided limits representing the concentration immediately before and immediately after the injection.

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