Answer
$\infty$
Work Step by Step
\begin{aligned}
\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{3}+x}-x^{\frac{3}{2}}\right) &=\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{3}+x}-x^{\frac{3}{2}}\right) \times \frac{\sqrt{9 x^{3}+x}+x^{\frac{3}{2}}}{\sqrt{9 x^{3}+x}+x^{\frac{3}{2}}}\\
&=\lim _{x \rightarrow \infty} \frac{9 x^{3}+x-x^{3}}{\sqrt{9 x^{3}+x}+x^{\frac{3}{2}}}\\
&=\lim _{x \rightarrow \infty} \frac{8 x^{3}+x}{\sqrt{9 x^{3}+x}+x^{\frac{3}{2}}}\\
&=\lim _{x \rightarrow \infty} \frac{x^{3}\left(8+\frac{1}{x^{2}}\right)}{\sqrt{x^{3}\left(9+\frac{1}{x^{2}}\right)}+x^{\frac{3}{2}}}\\
&=\lim _{x \rightarrow \infty} \frac{x^{3}\left(8+\frac{1}{x^{2}}\right)}{x^{\frac{3}{2}} \sqrt{\left(9+\frac{1}{x^{2}}\right)}+x^{\frac{3}{2}}}\\
&=\lim _{x \rightarrow \infty} \frac{x^{3}\left(8+\frac{1}{x^{2}}\right)}{x^{\frac{3}{2}}(\sqrt{\left(9+\frac{1}{x^{2}}\right)}+1)}\\
&=\lim _{x \rightarrow \infty} \frac{x^{\frac{3}{2}}\left(8+\frac{1}{x^{2}}\right)}{(\sqrt{\left(9+\frac{1}{x^{2}}\right)}+1)}\\
&=\frac{\infty(8+0)}{\sqrt{9+0}+1}=\frac{\infty}{3+1}=\infty
\end{aligned}
