Concentration of Drug in the Bloodstream The concentration of a certain drug in patient's bloodstream (t) hr after injection is given by:
C(t) = (0.2t) / (t^2+1)
mg/cm^3 Evaluate lim x-> inf C(t) and interpret your result.
as t gets huge, t^2+1 becomes just t^2
0.2t/t^2 = 0.2/t
and as t becomes huge, that vanishes
it just means the the body metabolizes any medication you provide, so the effects diminish.
To evaluate lim t-> inf C(t), we need to find the limiting value of C(t) as t approaches infinity.
To compute the limit, let's simplify the expression for C(t):
C(t) = (0.2t) / (t^2 + 1)
As t approaches infinity, the terms t^2 + 1 become negligible compared to t^2. Hence, we can ignore the 1 in the denominator.
C(t) ≈ (0.2t) / t^2
Simplifying further, we cancel out a factor of t from the numerator and denominator:
C(t) ≈ 0.2 / t
Now, as t approaches infinity, the value of 0.2/t becomes arbitrarily small. In other words, the concentration of the drug in the bloodstream approaches zero as time goes to infinity.
Therefore, we can interpret the result as the drug concentration becoming negligible or almost non-existent in the patient's bloodstream after a long time.
To find the limit of C(t) as t approaches infinity, we can substitute infinity into the equation:
lim t->∞ (0.2t) / (t^2+1)
We can simplify this equation by dividing every term by t^2:
lim t->∞ (0.2/t) / (1 + 1/t^2)
As t approaches infinity, both 0.2/t and 1/t^2 go towards zero, so we get:
lim t->∞ (0.2/t) / 1
= (0.2 / ∞) / 1
= 0 / 1
= 0
The limit of C(t) as t approaches infinity is equal to zero.
Interpretation:
This limit tells us that as time goes towards infinity (or very large values), the concentration of the drug in the patient's bloodstream approaches zero. In practical terms, it means that the drug is gradually getting eliminated from the patient's bloodstream and the concentration becomes negligible over time.