Question

A patient receives a 150-mg injection of a drug every 4 hours. The graphs shows the amount \(f\left( t \right)\) of the drug in the bloodstream after t hours. Find\(\mathop {{\bf{lim}}}\limits_{t \to {\bf{1}}{{\bf{2}}^ - }} f\left( t \right)\) and \(\mathop {{\bf{lim}}}\limits_{t \to {\bf{1}}{{\bf{2}}^ + }} f\left( t \right)\).

And explain the significance of these one-sided limits.

Short answer

150 mg and 300 mg

The left-hand side limit shows the amount of drug before the fourth injection. Similarly, the right-hand side limit shows the amount of drug after the fourth injection. The limits show an abrupt change in the amount of drugs.

Step-by-step solution

Step 1:Find the values of the limits

From the graph, it can be observed that as shown below:

\(\mathop {\lim }\limits_{t \to {{12}^ - }} f\left( t \right) = 150\;{\rm{mg}}\)(when t is approaching to 12 from the left-hand side.)

And,

\(\mathop {\lim }\limits_{t \to {{12}^ + }} f\left( t \right) = 300\;{\rm{mg}}\) (when t is approaching to 12 from right-hand side.)

Write an interpretation of results

The difference in values of limits at \(t = 12\) represents that there is an abrupt change in the amount of drug in bloodstream.

Theleft-hand side limit shows the amount of drug before the fourth injection.

Similarly, theright-hand side limit shows the amount of drug after the fourth injection.