Série 2 \( \begin{array}{lll}\text { aa) } f(x)=\frac{x}{1+x^{2}} & \text { ae) } f(x)=\frac{x}{(x-1)(x+2)^{2}} & \text { ai) } f(x)=|x-1| \\ \text { ab) } f(x)=x-\frac{1}{2 x} & \text { af) } f(x)=-\sqrt{x-3} & \text { aj) } f(x)=2 x+\frac{3}{x+4} \\ \text { ac) } f(x)=\frac{x^{3}+1}{x^{2}-1} & \text { ag) } f(x)=\frac{5 x}{x^{2}+6 x+9} & \text { ak) } f(x)=\frac{2 x}{\sqrt[3]{x^{2}-4}} \\ \text { ad) } f(x)=\sqrt[3]{x^{2}+3 x} & \text { ah) } f(x)=\frac{\sqrt{x^{2}-5 x+6}}{2 x-5} & \text { al) } f(x)=\frac{\sqrt{1-x}}{x^{3}+x}\end{array} \)

Ah, the world of functions! It’s like a colorful playground where each function behaves differently, and hidden treasures await in analysis! Imagine, for a moment, the function \( f(x) = \frac{x^3 + 1}{x^2 - 1} \). It’s not just a formula; it's a gateway! By decomposing this rational function via polynomial long division, you can unveil its asymptotic behavior, which plays a crucial role in determining limits and integrals. Dive into its graph, and you’ll see how it behaves as \( x \) approaches the roots of the denominator – those undefined moments where the party stops! Now, let’s talk about real-world application: the function \( f(x) = \frac{x}{(x-1)(x+2)^2} \) is not just math on paper. This function’s behavior, especially near its vertical asymptotes and zeroes, can model real-life scenarios like fluid flow in networks or even the dynamics of populations under certain constraints. By analyzing the discontinuities, you can predict stability scenarios in various systems—perfect for environmental scientists or engineers looking to optimize solutions!