Answer
$$\frac{5}{2}$$
Work Step by Step
\begin{aligned}
\lim _{x \rightarrow \infty}(\sqrt{x^{2}+3 x}-\sqrt{x^{2}-2 x}) &=\lim _{x \rightarrow \infty}[\sqrt{x^{2}+3 x}-\sqrt{x^{2}-2 x}] \cdot\left[\frac{\sqrt{x^{2}+3 x}+\sqrt{x^{2}-2 x}}{\sqrt{x^{2}+3 x}+\sqrt{x^{2}-2 x}}\right]\\
&=\lim _{x \rightarrow \infty} \frac{\left(x^{2}+3 x\right)-\left(x^{2}-2 x\right)}{\sqrt{x^{2}+3 x}+\sqrt{x^{2}-2 x}} \\
&=\lim _{x \rightarrow \infty} \frac{5 x}{\sqrt{x^{2}+3 x}+\sqrt{x^{2}-2 x}}\\
&=\lim _{x \rightarrow \infty} \frac{5}{\sqrt{1+\frac{3}{x}}+\sqrt{1-\frac{2}{x}}}\\
&=\frac{5}{1+1}=\frac{5}{1+1}\\
&=\frac{5}{2}
\end{aligned}
