91Ó°ÊÓ

Rational functions are ratios of polynomial functions, where one polynomial is divided by another.
The general form of a rational function is \[\begin{equation}R(x) = \frac{p(x)}{q(x)}\end{equation}\]where both p(x) and q(x) are polynomials and q(x) is not the zero polynomial. In the exercise, the concentration function \[\begin{equation}C(t)=\frac{10 t}{1+t^{2}}\end{equation}\]is a rational function of time \[\begin{equation}t\end{equation}\]since both the numerator \[\begin{equation}10t\end{equation}\]and the denominator \[\begin{equation}1+t^{2}\end{equation}\]are polynomials in terms of t.
Rational functions often describe real-world situations where there is a rate of change that is contingent upon the current value of the variable, such as the concentration of the drug in the bloodstream over time in our exercise. They can show growth or decline and can have varying rates throughout their domain.

A horizontal asymptote is a horizontal line that a function approaches as the independent variable, usually x or in our case t, tends towards infinity or negative infinity.
In the context of rational functions, we determine the horizontal asymptote by comparing the degrees of the polynomials in the numerator and denominator.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0.
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

For our drug concentration function \[\begin{equation}C(t)=\frac{10 t}{1+t^{2}}\end{equation}\]the numerator and denominator both have degrees of 1, hence the function will have a horizontal asymptote. This line, which we find not by intersection but by limits, represents a boundary that the output value of the function will never exceed, indicating the maximum possible concentration of the drug in the bloodstream in this case.

Limits are fundamental in calculus and help describe the behavior of functions as inputs approach a certain value. The limit of a function f(x) as x approaches a value a essentially answers the question, 'What value does f(x) get closer to as x gets arbitrarily close to a?' When solving for horizontal asymptotes mathematically, we make use of limits to infinity.
The notation \[\begin{equation}\lim_{{x \to a}} f(x)\end{equation}\]is used to represent the limit of f(x) as x approaches a. In the example of drug concentration in a bloodstream, to find the asymptotic concentration as time goes to infinity, we calculate it using the limit \[\begin{equation}\lim_{{t \to \infty}} \frac{10t}{1 + t^{2}} = 10\end{equation}\]This helps us understand how the drug will behave over a long period and is essential for determining long-term dosing and therapeutic levels in medical applications.