Answer
$$\lim\limits_{x\to-\infty}(\sqrt{4x^2+3x}+2x)=-\frac{3}{4}$$
Work Step by Step
$$A=\lim\limits_{x\to-\infty}(\sqrt{4x^2+3x}+2x)$$$$A=\lim\limits_{x\to-\infty}\Bigg[(\sqrt{4x^2+3x}+2x)\times\frac{\sqrt{4x^2+3x}-2x}{\sqrt{4x^2+3x}-2x}\Bigg]$$$$A=\lim\limits_{x\to-\infty}\frac{(4x^2+3x)-4x^2}{\sqrt{4x^2+3x}-2x}$$$$A=\lim\limits_{x\to-\infty}\frac{3x}{\sqrt{4x^2+3x}-2x}$$$$A=\lim\limits_{x\to-\infty}\frac{X}{Y}$$
Divide both numerator and denominator by $x$
Notice that as $x\to-\infty$, we have $x\lt0$. Therefore, $\sqrt{x^2}=|x|=-x$
- The numerator:
$X=\frac{3x}{x}=3$
- The denominator:
$Y=\frac{\sqrt{4x^2+3x}-2x}{x}=\frac{\sqrt{4x^2+3x}}{x}-\frac{2x}{x}=\frac{\sqrt{4x^2+3x}}{-\sqrt{x^2}}-2=-\sqrt{4+\frac{3}{x}}-2$
Therefore, $$A=\lim\limits_{x\to-\infty}\frac{3}{-\sqrt{4+\frac{3}{x}}-2}$$$$A=\frac{3}{-\sqrt{4+\lim\limits_{x\to-\infty}(\frac{3}{x})}-2}$$$$A=\frac{3}{-\sqrt{4+0}-2}$$$$A=-\frac{3}{4}$$
