Next, divide $ -2t $ by $ t $ to get $ -2 $. Multiply $ t + 2 $ by $ -2 $ to get $ -2t - 4 $. Subtract:

February 22, 2026

Next, divide $ -2t $ by $ t $ to get $ -2 $. Multiply $ t + 2 $ by $ -2 $ to get $ -2t - 4 $. Subtract:

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Question: A zoologist observes that the number of daily interactions between two species in a reserve is modeled by the function $ I(t) = \frac{t^2 + 4}{t + 2} $, where $ t $ is time in days since the start of the study. Find the slant asymptote of $ I(t) $ as $ t \to \infty $.
Solution: To find the slant asymptote of $ I(t) = \frac{t^2 + 4}{t + 2} $, we perform polynomial long division of $ t^2 + 4 $ by $ t + 2 $:
Divide $ t^2 $ by $ t $ to get $ t $. Multiply $ t + 2 $ by $ t $ to get $ t^2 + 2t $. Subtract:
I(t) = t - 2 + \frac{8}{t + 2}.
As $ t \to \infty $, the term $ \frac{8}{t + 2} \to 0 $, so the slant asymptote is:
\boxed{y = t - 2}

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