We are given the limit:

$\lim_{x \to \infty} \left( \frac{\sqrt[3]{x+1}}{x^2 + 3} \cdot \arctan(x) \right)$

Step-by-step solution:

1. Analyze the behavior of each term as $x \to \infty$:

   • For $\sqrt[3]{x+1}$, at large values of $x$, this expression behaves approximately like $\sqrt[3]{x}$, because $\frac{1}{x}$ becomes negligible. Hence, $\sqrt[3]{x+1} \sim x^{1/3}$.
   • For the denominator $x^2 + 3$, as $x \to \infty$, this term behaves like $x^2$.
   • The arctangent term, $\arctan(x)$, tends to $\frac{\pi}{2}$ as $x \to \infty$.

2. Rewrite the expression in terms of dominant terms as $x \to \infty$: $\frac{\sqrt[3]{x+1}}{x^2 + 3} \sim \frac{x^{1/3}}{x^2} = \frac{1}{x^{5/3}}$ Thus, the original expression for large $x$ can be approximated by: $\lim_{x \to \infty} \frac{1}{x^{5/3}} \cdot \arctan(x) \sim \lim_{x \to \infty} \frac{\pi/2}{x^{5/3}}$

3. Final Limit: Since $\frac{\pi/2}{x^{5/3}} \to 0$ as $x \to \infty$, the limit is:

$\boxed{0}$

Would you like any further details or clarifications?

Here are 5 related questions to explore this topic further:

1. How do we handle limits involving multiple functions like this one?
2. What is the growth rate of the cubic root compared to polynomial expressions?
3. How does the arctangent function behave for large $x$?
4. How can L'Hopital's rule be applied in complex limits?
5. What other techniques can simplify limits at infinity?

Tip: When analyzing limits involving multiple terms, always focus on how each part of the expression behaves as $x \to \infty$, and determine the dominant term.