?Find the limit or show that it does not exist.\(\lim _{t \rightarrow-\infty} \frac{3

Step 1 of 2

Consider the given function is,

\(\lim _{t \rightarrow \infty} \frac{3 t^{2}+1}{t^{3}-4 t+1}\)

Divide both the numerator and the denominator by the highest power of t of the denominator.

\(\begin{aligned}
\lim _{t \rightarrow \infty} \frac{3 t^{2}+1}{t^{3}-4 t+1} & =\lim _{t \rightarrow \infty} \frac{\frac{3 t^{2}}{t^{3}}+\frac{1}{t^{3}}}{t^{3}-\frac{4 t}{t^{3}}+\frac{1}{t^{3}}} \\
& =\lim _{t \rightarrow \infty} \frac{\frac{3}{t}+\frac{1}{t^{3}}}{1-\frac{4}{t^{2}}+\frac{1}{t^{3}}}
\end{aligned}\)