Answer
$0$
Work Step by Step
$\displaystyle \lim_{x\rightarrow\infty}\frac{4x^{3}}{x^{4}+3}= \displaystyle \lim_{x\rightarrow\infty}[\frac{4x^{3}\div x^{4}}{(x^{4}+3)\div x^{4}}$
$=\displaystyle \lim_{x\rightarrow\infty}\frac{\frac{4}{x}}{1+\frac{3}{x^{4}}}$
... the terms $\displaystyle \frac{4}{x}$ and $\displaystyle \frac{3}{x^{4}}$ both approach 0 when $ x\rightarrow\infty$
$=\displaystyle \frac{0}{1+0}=0$
