$\lim\limits_{x\to-\infty}\sqrt{4x^2+3x}+2x$

Question

I want to solve this limit:

$$\lim_{x\to-\infty}\sqrt{4x^2+3x}+2x$$

My try was to multiply by the conjugate, which gave me

$$\lim_{x\to-\infty}\dfrac{3x}{\sqrt{4x^2+3x}-2x}$$

But then factoring $x$ out of the denominator and cancelling with the $x$ at the numerator gives me

$\lim\limits_{x\to-\infty}\dfrac{3}{(4+\frac{3}{x})^{1/2}-2}$

The problem is, when I evaluate this limit, I get $\frac{3}{0}$, but my book says the limit should be $\frac{3}{4}$.

Can anyone see where I made my mistake?

Answer 1

When factoring $x$ out you should check the sign of $x$. You might a mistake when treating $\sqrt{4x^2+3x}$. Our $x$ is negative so $\sqrt{4x^2+3x}$ is equal to $-x\sqrt{4-3/x}$.

Answer 2

Note that $x$ is negative (as $x \to -\infty$). So, $$ \frac{\sqrt{4x^2 + 3x}}{x} = \frac{\sqrt{4x^2 + 3x}}{-\sqrt{x^2}} = -\sqrt{4 + \frac 3x} $$