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

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


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\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} }
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=& \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) }
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=& \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}}
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=& \frac{0 + 0}{1 + 0 - 0}
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=& \frac{0}{1}
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=& 0

\end{aligned}
\end{equation}
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