In your analysis of
\begin{equation*}
g(t) = \frac{2 t^5 - 3 t^3 + t + 4}{3 t^4 + 4 t^3 + 6}
\end{equation*}
\begin{equation*}
g(t) \approx \frac{2 t}{3}
\end{equation*}
for large values of \(t\text{,}\) whether positive or negative. From this we can conclude that
\begin{equation*}
g(t) \to \infty \qquad \text{as} \qquad t \to \infty
\end{equation*}
and
\begin{equation*}
g(t) \to -\infty \qquad \text{as} \qquad t \to -\infty \text{.}
\end{equation*}
However, the expression
\begin{equation*}
g(t) \approx \frac{2}{3} t
\end{equation*}
is actually more informative about exactly how \(g(t) \to \pm \infty\text{;}\) in particular, it says that \(g(t)\) is essentially linear for large values of \(t\text{,}\) growing at approximately constant rate \(2/3\text{.}\)
However, there may be an “offset” to how it grows linearly. To see this, let’s investigate more closely how the numerator and denominator polynomials relate. In the numerator, the only term that matters is the \(2 t^5\) since the other terms are “small” compared to the \(3 t^4\) in the denominator:
\begin{align*}
g(t) \amp = \frac{2 t^5 - 3 t^3 + t + 4}{3 t^4 + 4 t^3 + 6} \\
\amp = \frac{2 t^5}{3 t^4 + 4 t^3 + 6} + \frac{- 3 t^3 + t + 4}{3 t^4 + 4 t^3 + 6} \\
\amp \approx \frac{2 t^5}{3 t^4 + 4 t^3 + 6} \text{,}
\end{align*}
where the approximately equal is true for large \(t\text{.}\) Now let’s make the numerator look more like the denominator.
\begin{align*}
g(t) \amp \approx \frac{2 t^5}{3 t^4 + 4 t^3 + 6} \\
\amp = 2 t \cdot \frac{t^4}{3 t^4 + 4 t^3 + 6} \\
\amp = \frac{2}{3} \, t \cdot \frac{3 t^4}{3 t^4 + 4 t^3 + 6} \\
\amp = \frac{2}{3} \, t \cdot \frac{3 t^4 + 4 t^3 + 6 - (4 t^3 + 6)}{3 t^4 + 4 t^3 + 6} \text{.}
\end{align*}
In this last step, we have added and then immediately subtracted some terms to make the numerator look even more like the denominator, while maintaining equality with the previous expression. Continuing,
\begin{align*}
g(t) \amp \approx \frac{2}{3} \, t \cdot \frac{3 t^4 + 4 t^3 + 6 - (4 t^3 + 6)}{3 t^4 + 4 t^3 + 6} \\
\amp =
\frac{2}{3} \, t \cdot \frac{3 t^4 + 4 t^3 + 6}{3 t^4 + 4 t^3 + 6}
- \frac{2}{3} \, t \cdot \frac{4 t^3 + 6}{3 t^4 + 4 t^3 + 6}\\
\amp =
\frac{2}{3} \, t \cdot 1
- \frac{2}{3} \cdot \frac{4 t^4 + 6 t}{3 t^4 + 4 t^3 + 6}\text{.}
\end{align*}
Looking at the dominant terms in the remaining rational expression, we have
\begin{align*}
g(t) \amp \approx
\frac{2}{3} \, t
- \frac{2}{3} \cdot \frac{4}{3}\\
\amp = \frac{2}{3} \, t - \frac{8}{9} \text{.}
\end{align*}
We conclude that for large \(t\text{,}\) the graph of \(g(t)\) will look like the line
\begin{equation*}
y = \frac{2}{3} \, t - \frac{8}{9} \text{.}
\end{equation*}
