The task in the image is to find the asymptotes of the following functions:

a) $y = \sqrt[3]{1 + x^3}$

b) $y = \ln(1 + e^{-x})$

c) $y = \frac{x^3 \arccot(x)}{1 + x^2}$

Let's analyze each one to find the asymptotes.

a) $y = \sqrt[3]{1 + x^3}$

• As $x \to \infty$: $y \sim \sqrt[3]{x^3} = x.$ So, the function behaves like $y = x$, implying there is a slant asymptote with equation $y = x$ as $x \to \infty$.

• As $x \to -\infty$: $y \sim \sqrt[3]{x^3} = x.$ Similarly, $y \sim x$, so the asymptote for $x \to -\infty$ is also $y = x$.

Thus, the asymptote for large $x$ values is $y = x$.

b) $y = \ln(1 + e^{-x})$

• As $x \to \infty$: $e^{-x} \to 0 \quad \Rightarrow \quad y \sim \ln(1) = 0.$ Therefore, there is a horizontal asymptote at $y = 0$ as $x \to \infty$.

• As $x \to -\infty$: $e^{-x} \to \infty \quad \Rightarrow \quad y \sim \ln(e^{-x}) = -x.$ So the asymptote as $x \to -\infty$ is a slant asymptote with equation $y = -x$.

c) $y = \frac{x^3 \arccot(x)}{1 + x^2}$

• As $x \to \infty$: $\arccot(x) \sim \frac{1}{x} \quad \Rightarrow \quad y \sim \frac{x^3 \cdot \frac{1}{x}}{1 + x^2} = \frac{x^2}{x^2 + 1} \sim 1.$ Therefore, there is a horizontal asymptote at $y = 1$ as $x \to \infty$.

• As $x \to -\infty$: Since $\arccot(x) \sim \pi - \frac{1}{x}$, we get: $y \sim \frac{x^3 (\pi - \frac{1}{x})}{1 + x^2} \sim \frac{\pi x^3}{x^2} = \pi x.$ So as $x \to -\infty$, there is a slant asymptote at $y = \pi x$.

Let me know if you'd like further details on any of the steps. Here are some follow-up questions:

1. What are the general conditions for the existence of horizontal asymptotes?
2. How would you approach finding vertical asymptotes for a rational function?
3. Can a function have more than one horizontal asymptote?
4. What is the significance of the behavior of $\arccot(x)$ as $x \to \pm\infty$?
5. How do we derive slant asymptotes when the degree of the numerator exceeds that of the denominator?

Tip: Slant asymptotes occur when the degree of the numerator is exactly one more than the denominator in rational functions.