<s>[INST] 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.
 [/INST] Hint
Setting limit to $0^+$ often helps me in limit
Set $-\dfrac1x=h\implies h\to0^+$
$$\sqrt{4x^6+3}=\sqrt{\dfrac{4+3h^6}{h^6}}=\dfrac{\sqrt{4+3h^6}}{|h^3|}$$
As $h>0,|h^3|=|h|^3=(+h)^3$
But $x^3=\left(-\dfrac1h\right)^3=-\dfrac1{h^3}$
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