Answer
L’Hôpital’s Rule does not apply and the limit is $0$.
Work Step by Step
Since we have
$$
\lim _{x \rightarrow 0} \frac{\sin 4x}{x^{2}+3x+1}=\frac{0}{1}=0.
$$
then L’Hôpital’s Rule does not apply and the limit is $0$.
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