\begin{tabular}{l} Question \\ Using L'Hospital's rule, determine whether \( f(x) \) or \( g(x) \) given below grows at a faster rate. \\ \( \qquad f(x)=-3 x^{2}-x-1 \) \\ \( g(x)=-x^{3}-5 x-2 \) \\ Select the correct answer below: \\ f(x) grows at a faster rate. \\ g(x) grows at a faster rate. \\ f(x) and \( g(x) \) grow at the same rate. \\ There is not enough information to determine which function grows at a faster rate. \\ \hline\end{tabular}

Proporre un esempio di funzione \( g(x) \) tale che \[ \lim _{x \rightarrow+\infty} \frac{g(x)}{3 \log ^{2} x+1}=+\infty \]

\( \lim _ { n \rightarrow + \infty } \frac { n ^ { 5 } + \sin n } { 3 ( \arctan n ) n ^ { 5 } } \)

MAS182 - Applied Mathematics 13. Final Examination - S1 2021 The water content \( W(t) \) (measured in grams) in the cheese is governed by the differential equation \[ \frac{d W}{d t}=-(W-100)^{2} \] with initial condition \( W(0)=200 \mathrm{~g} \) and time is measured in hours.

5. [12 marks] Using calculus and showing all your working, sketch the function \[ f(x)=\frac{x}{x^{2}-4} \] noting the domain of \( f(x) \), any intercepts or asymptotes, symmetry, regions where the function is increasing or decreasing and any relative extrema. Note: You don't have to consider the second derivative and concavity.

asymptotes of \( f \).