The problem you uploaded asks to evaluate the following limit:

$\lim_{t \to \infty} \frac{t^3 + t}{5 - t^2 - 2t^3}$

Step-by-Step Solution:

1. Simplify the expression: We need to identify the dominant terms as $t \to \infty$. The highest power of $t$ in the numerator is $t^3$, and in the denominator, the highest power of $t$ is also $t^3$. Therefore, as $t \to \infty$, the other terms (like $t$ and $t^2$) become negligible.

2. Factor out $t^3$ from both the numerator and the denominator to make the dominant terms clearer: [ \frac{t^3 + t}{5 - t^2 - 2t^3} = \frac{t^3(1 + \frac{1}{t^2})}{t^3(-2 +