Trouble finding limits at $-\infty$. I'm being asked to evaluate $\lim\limits_{x\rightarrow\infty}f(x)$ and $\lim\limits_{x\rightarrow -\infty}f(x)$ for the following funtion: $f(x)=\frac{4x^3+2}{x^3+\sqrt{4x^6+3}}$.
For the first limit I proceed as follows:
$\begin{align*}
\lim\limits_{x\rightarrow\infty}f(x)&=\lim\limits_{x\rightarrow\infty}\frac{4x^3+2}{x^3+\sqrt{4x^6+3}}\\&=\lim\limits_{x\rightarrow\infty}\frac{(4+\frac{2}{x^3})}{(1+\sqrt{4+\frac{3}{x^6})}}\\&=\frac{4}{3}
\end{align*}$
This is the horizontal asymptote as $x\rightarrow\infty$, however, when I work through it again for $-\infty$, I get the same answer, but I know that as $x\rightarrow -\infty$, $f(x)$ approaches $-4$. Obviously there is some hole in my understanding. Any clarification would be greatly appreiciated! Thank you.

When $x<0$ we have $x^{-3} \sqrt{4x^6+3} = -\sqrt{x^{-6}(4x^6+3)} = - \sqrt{4 + 3x^{-6}}$.
