Answer
$$\lim _{t \rightarrow \infty} \frac{e^{2 t}}{2 t}=\infty$$
Work Step by Step
Given $$\lim _{t \rightarrow \infty} \frac{e^{2 t}}{2 t}$$
using the Limit Rules and replacement, leads to the indeterminate form $$\lim _{t \rightarrow \infty} \frac{e^{2 t}}{2 t}=\frac{\infty}{\infty}$$
Applying L'Hôpital's Rule
\begin{align*}
\lim _{t \rightarrow \infty} \frac{e^{2 t}}{2 t}&=\lim _{t \rightarrow \infty} \frac{2e^{2 t}}{2 }\\
&=\lim _{t \rightarrow \infty} e^{2 t} \\
&=\lim _{t \rightarrow \infty} e^{\infty}\\
&=\infty
\end{align*}
