The task asks to find the limit of the sequence:

$x_n = \frac{2 - 3n^3}{4 + 5n^2}.$

Let’s calculate the limit as $n \to \infty$.

Solution:

1. Identify the dominant terms in the numerator and denominator:

   • In the numerator $2 - 3n^3$, the dominant term is $-3n^3$.
   • In the denominator $4 + 5n^2$, the dominant term is $5n^2$.

2. Divide all terms by $n^3$ (the highest power in the numerator): $x_n = \frac{\frac{2}{n^3} - 3}{\frac{4}{n^3} + \frac{5}{n}}.$

3. Simplify as $n \to \infty$:

   • As $n \to \infty$, terms like $\frac{2}{n^3}$ and $\frac{4}{n^3}$ approach 0.
   • The sequence simplifies to: $x_n \to \frac{-3}{0 + 0}.$

Since the denominator blows up while the numerator stays finite, the limit does not exist as finite. The dominant nature concludes this sequence is divergent.

Would you like further steps broken-down?