Final answer:
As time t increases, the concentration of the medication in the bloodstream, given by C(t)=3t/(t² + 2), decreases towards zero, showing that the medication concentration diminishes over time.
Step-by-step explanation:
The concentration of a medication in a patient's bloodstream t hours after the injection is represented by C(t)=3t/(t² + 2). To understand what happens to the concentration as time t increases, we can analyze the behavior of the function as t approaches infinity.
For large values of t, the t² term in the denominator will dominate over the constant '2', causing the value of t² + 2 to be approximately equal to t². Therefore, the concentration function can be approximated as C(t) ≈ 3t/t² = 3/t when t is large. This simplification shows that as t increases, the concentration C(t) decreases towards zero, indicating that the medication is being metabolized and its concentration in the bloodstream is diminishing over time.