Limits

$\begin{aligned} \lim_{x \to \infty} \left(\sqrt{x^2+3x}-x\right) &= \lim_{x \to \infty} \left[ \left(\sqrt{x^2+3x}-x\right) \left(\frac{\sqrt{x^2+3x}+x}{\sqrt{x^2+3x}+x}\right) \right] & \small \color{#3D99F6} \text{Multiply with conjugate} \\ &= \lim_{x \to \infty} \left(\frac{x^2+3x-x^2}{\sqrt{x^2+3x}+x}\right) \\ &= \lim_{x \to \infty} \left(\frac{3x}{\sqrt{x^2+3x}+x}\right) \\ &= \lim_{x \to \infty} \left[\left(\frac{3x}{\sqrt{x^2+3x}+x}\right) \left(\frac{\frac{1}{x}}{\frac{1}{x}}\right)\right] & \small \color{#3D99F6} \text{Eliminate } x \text{ from the numerator} \\ &= \lim_{x \to \infty} \left(\frac{3}{\sqrt{\frac{1}{x^2}\left(x^2+3x\right)}+1}\right) \\ &= \lim_{x \to \infty} \left(\frac{3}{\sqrt{1+\frac{3}{x}}+1}\right) \\ &= \frac{3}{\sqrt{1+0}+1} \\ &= \frac{3}{2} \ _\square \end{aligned}$