QuestionFind the horizontal asymptote of $C(t)=\frac{t}{7t^{2}+8}$ . What does $C(t)$ approach as $t$ increases?

Studdy Solution

STEP 1

Assumptions1. The concentration $C$ of the drug in the patient's bloodstream is given by the function $C(t)=\frac{t}{7 t^{}+8}$ . . We are asked to find the horizontal asymptote of $C(t)$ , which is the value that $C(t)$ approaches as $t$ goes to infinity.

STEP 2

To find the horizontal asymptote of a rational function, we need to look at the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$ .
In our case, the degree of the numerator (which is $1$ ) is less than the degree of the denominator (which is $2$ ). Therefore, the horizontal asymptote is $y=0$ .

STEP 3

Now, let's determine what happens to the concentration of the drug as $t$ increases. For this, we need to find the limit of $C(t)$ as $t$ goes to infinity.
$\lim{{t \to \infty}} C(t) = \lim{{t \to \infty}} \frac{t}{7 t^{2}+8}$

STEP 4

To calculate this limit, we can use the rule that the limit of a rational function as $t$ goes to infinity is equal to the ratio of the leading coefficients if the degrees of the numerator and the denominator are equal. If the degree of the numerator is less than the degree of the denominator, the limit is $0$ .
In our case, the degree of the numerator is less than the degree of the denominator. Therefore, the limit is $0$ .
$\lim{{t \to \infty}} C(t) =0$

STEP 5

So, as $t$ increases, the concentration of the drug in the patient's bloodstream approaches $0$ . This means that the drug is gradually eliminated from the patient's bloodstream.
The horizontal asymptote of $C(t)$ is $y=0$ and as $t$ increases, $C(t)$ approaches $0$ .

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