Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 14

Determine $\displaystyle \lim_{t \to -\infty} \frac{t^2 + 2}{t^3 + t^2 - 1}$


$\begin{equation} \begin{aligned} \lim_{t \to -\infty} \frac{t^2 + 2}{t^3 + t^2 - 1} \cdot \frac{\displaystyle \frac{1}{t^3}}{\displaystyle \frac{1}{t^3}} =& \lim_{t \to -\infty} \frac{\displaystyle \frac{t^2}{t^3} + \frac{2}{t^3}}{\displaystyle \frac{\cancel{t^3}}{\cancel{t^3}} + \frac{t^2}{t^3} - \frac{1}{t^3} } \\ \\ =& \frac{\displaystyle \lim_{t \to -\infty} \left( \frac{1}{t} + \frac{2}{t^3}\right) }{\displaystyle \lim_{t \to - \infty} \left( 1 + \frac{1}{t} - \frac{1}{t^3} \right) } \\ \\ =& \frac{\displaystyle \lim_{t \to - \infty} \frac{1}{t} + \lim_{t \to - \infty} \frac{2}{t^3} }{\displaystyle 1 + \lim_{t \to - \infty} \frac{1}{t} - \lim_{t \to - \infty} \frac{1}{t^3}} \\ \\ =& \frac{0 + 0}{1 + 0 - 0} \\ \\ =& \frac{0}{1} \\ \\ =& 0 \end{aligned} \end{equation}$