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When exploring the asymptotic behavior of rational functions like \(c(t)=\frac{30 t}{t^{2}+2}\), we seek to understand how the function behaves as \(t\) becomes extremely large or extremely small. Asymptotic behavior for this function is primarily determined by horizontal asymptotes, since there are no vertical asymptotes in this scenario.

A horizontal asymptote occurs when the outputs of a function approach a constant value as \(t\) approaches infinity. For \(c(t)\), as \(t\) increases, the quantity \(t^2\) in the denominator grows much faster than \(30t\) in the numerator. This growth causes the entire expression to approach zero, as calculated using L'Hôpital's Rule.

Understanding this behavior is crucial in real-world contexts, such as drug concentration in blood, where we deduce that over time, the drug's effectiveness diminishes and approaches a baseline level, which is zero in this case. Knowing this can be vital for determining dosing schedules in medical treatments.

Graphing functions, especially rational functions like \(c(t)=\frac{30 t}{t^{2}+2}\), allows us to visually interpret and analyze the behavior of real-world scenarios such as drug concentration over time. To graph this function, several key features must be identified: intercepts, asymptotes, and general behavior over its domain.

• Intercepts: For \(c(t)\), the y-intercept can be found by setting \(t=0\). This calculation provides \(c(0) = 0\). There are no x-intercepts as the function never crosses the x-axis meaning there's no solution where the function equals zero unless \(t=0\).
• Asymptotes: Understanding there are no vertical asymptotes because \(t^2+2\) never becomes zero highlights how the function's domain is unrestricted for all real numbers. The horizontal asymptote at \(y=0\) impacts the function's long-term behavior.

To graph, we plot from the origin, observe the rise to a peak, then a gradual decline approaching the horizontal asymptote. Such visual tools are critical in making predictions and understanding trends in data like concentration changes.

Calculating limits helps us understand the behavior of a function as the variable approaches a particular value (like infinity). For \(c(t)=\frac{30 t}{t^{2}+2}\), the limit calculation as \(t\to \infty\) is essential to predicting the concentration's eventual value. This is achieved using L'Hôpital's Rule, a calculative technique for limits of indeterminate forms.

By differentiating the numerator and denominator separately, we find:\[\lim_{{t \to \infty}} \frac{30}{2t} = 0\]This result indicates that the concentration decreases to zero over time.

Calculating limits in this context clarifies the final outcome of processes, like drug concentration, in complex systems. It reflects why certain dosages cannot sustain long-term effectiveness and why repeated doses might be necessary to maintain therapeutic concentration levels. This understanding is foundational in fields involving dynamic concentration changes such as pharmacokinetics and other applied sciences.