Answer
Indeterminate form is $\dfrac{ \infty}{ \infty}$ and$$\lim _{x \rightarrow \infty} \frac{3 x^{4}}{5 x^{3}+6} = \infty$$
Work Step by Step
Given $$\lim _{x \rightarrow \infty} \frac{3 x^{4}}{5 x^{3}+6}$$
using the method of replacement
\begin{align*}
\lim _{x \rightarrow \infty} \frac{3 x^{4}}{5 x^{3}+6}&=\frac{\infty}{\infty}
\end{align*}
This is an indeterminate form, then applying L'Hôpital's Rule, we
get
\begin{align*}
\lim _{x \rightarrow \infty} \frac{3 x^{4}}{5 x^{3}+6}&=\lim _{x \rightarrow \infty} \frac{12 x^{3}}{15 x^{2}+6} \ \ \text{apply L'Hôpital's Rule}\\
&=\lim _{x \rightarrow \infty} \frac{36 x^{2}}{30 x} \ \ \text{apply L'Hôpital's Rule}\\
&=\lim _{x \rightarrow \infty} \frac{72x}{30 }\\
&=\lim _{x \rightarrow \infty} \frac{ \infty}{30 }\\
&= \infty
\end{align*}
