To determine the limits at infinity for the given rational function and identify any horizontal asymptotes, let's analyze the behavior of the function as xx approaches both $\backslash$infty and $$$-\backslash$infty$.$

The given function is:

f$($x$)=4$x$3+2$x$3+4$x$6+2$f$($x$)=\backslash$frac$\{4$x$^3+2\}\{$x$^3+\backslash$sqrt$\{4$x$^6+2\}\}$

Step $1$: Simplify the Function

To simplify, we can factor out x$6$x$^6$ from inside the square root in the denominator:

$4$x$6+2=$x$6(4+2$x$6)=$x$34+2$x$6\backslash$sqrt$\{4$x$^6+2\}=\backslash$sqrt$\{$x$^6(4+\backslash$frac$\{2\}\{$x$^6\})\}=$ x$^3\backslash$sqrt$\{4+\backslash$frac$\{2\}\{$x$^6\}\}$

So the function becomes:

f$($x$)=4$x$3+2$x$3+$x$34+2$x$6$f$($x$)=\backslash$frac$\{4$x$^3+2\}\{$x$^3+$ x$^3\backslash$sqrt$\{4+\backslash$frac$\{2\}\{$x$^6\}\}\}$

Factor x$3$x$^3$ out from the denominator:

f$($x$)=4$x$3+2$x$3(1+4+2$x$6)=4$x$3+2$x$3(1+2)$f$($x$)=\backslash$frac$\{4$x$^3+2\}\{$x$^3(1+\backslash$sqrt$\{4+\backslash$frac$\{2\}\{$x$^6\}\})\}=\backslash$frac$\{4$x$^3+2\}\{$x$^3(1+2)\}$

Simplify further:

f$($x$)=4$x$3+23$x$3=4+2$x$33=43+23$x$3$f$($x$)=\backslash$frac$\{4$x$^3+2\}\{3$x$^3\}=\backslash$frac$\{4+\backslash$frac$\{2\}\{$x$^3\}\}\{3\}=\backslash$frac$\{4\}\{3\}+\backslash$frac$\{2\}\{3$x$^3\}$

Step $2$: Evaluate the Limits

$1.$ limx$$f$($x$)\backslash$lim$\_\{\{$x $\backslash$to $\backslash$infty$\}\}$ f$($x$)$

As xx approaches $\backslash$infty$,$ the term $2$x$3\backslash$frac$\{2\}\{$x$^3\}$ approaches $0.$ Therefore, the expression simplifies to:

limx$$f$($x$)=43+0=43\backslash$lim$\_\{\{$x $\backslash$to $\backslash$infty$\}\}$ f$($x$)=\backslash$frac$\{4\}\{3\}+0=\backslash$frac$\{4\}\{3\}$

$2.$ limx$$f$($x$)\backslash$lim$\_\{\{$x $\backslash$to $-\backslash$infty$\}\}$ f$($x$)$

Similarly, as xx approaches $$$-\backslash$infty$,$ the term $2$x$3\backslash$frac$\{2\}\{$x$^3\}$ approaches $0.$ Therefore, the expression simplifies to:

limx$$f$($x$)=43+0=43\backslash$lim$\_\{\{$x $\backslash$to $-\backslash$infty$\}\}$ f$($x$)=\backslash$frac$\{4\}\{3\}+0=\backslash$frac$\{4\}\{3\}$

Step $3$: Determine Horizontal Asymptotes

Since both the limits as xx approaches $\backslash$infty and $$$-\backslash$infty yield the same value, the horizontal asymptote is:

y$=43$y $=\backslash$frac$\{4\}\{3\}$

Conclusion

limx$$f$($x$)=43\backslash$lim$\_\{\{$x $\backslash$to $\backslash$infty$\}\}$ f$($x$)=\backslash$frac$\{4\}\{3\}$

limx$$f$($x$)=43\backslash$lim$\_\{\{$x $\backslash$to $-\backslash$infty$\}\}$ f$($x$)=\backslash$frac$\{4\}\{3\}$

The horizontal asymptote of the function f$($x$)=4$x$3+2$x$3+4$x$6+2$f$($x$)=\backslash$frac$\{4$x$^3+2\}\{$x$^3+\backslash$sqrt$\{4$x$^6+2\}\}$ is y$=43$y $=\backslash$frac$\{4\}\{3\}.$