Answer
$$\frac{1}{2}$$
Work Step by Step
Given
$$ \lim _{t \rightarrow \infty} \frac{t^{3}-t+2}{(2 t-1)\left(t^{2}+t+1\right)}$$
Then
\begin{aligned} \lim _{t \rightarrow \infty}\left[\frac{t^{3}-t+2}{(2 t-1)\left(t^{2}+t+1\right)} \cdot \frac{\frac{1}{t} \cdot \frac{1}{t^{2}}}{\frac{1}{t} \cdot \frac{1}{t^{2}}}\right] &=\lim _{t \rightarrow \infty}\left[\frac{t^{3}-t+2}{(2 t-1)\left(t^{2}+t+1\right)} \cdot \frac{\frac{1}{t^{3}}}{\frac{1}{t} \cdot \frac{1}{t^{2}}}\right] \\ &=\lim _{t \rightarrow \infty}\left[\frac{1-1 / t^{2}+2 / t^{3}}{\frac{1}{t}(2 t-1) \cdot \frac{1}{t^{2}}\left(t^{2}+t+1\right)}\right] \\ &=\lim _{t \rightarrow \infty}\left[\frac{1-1 / t^{2}+2 / t^{3}}{(2-1 / t)\left(1+1 / t+1 / t^{2}\right)}\right] \\ &=\frac{1-0+0}{(2-0)(1+0+0)} \\ &=\frac{1}{2} \end{aligned}
