Rewrite \({x}^{3}-1\) in the form \({a}^{3}-{b}^{3}\), where \(a=x\) and \(b=1\).
\[l\imath mx->0{(arc\tan{(\frac{{x}^{2}-\sqrt{3}}{{x}^{3}-{1}^{3}})})}^{\frac{x}{\sin{2x}}}\]
Use Difference of Cubes : \({a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})\).
\[l\imath mx->0{(arc\tan{(\frac{{x}^{2}-\sqrt{3}}{(x-1)({x}^{2}+(x)(1)+{1}^{2})})})}^{\frac{x}{\sin{2x}}}\]
Simplify \({1}^{2}\) to \(1\).
\[l\imath mx->0{(arc\tan{(\frac{{x}^{2}-\sqrt{3}}{(x-1)({x}^{2}+x\times 1+1)})})}^{\frac{x}{\sin{2x}}}\]
Simplify \(x\times 1\) to \(x\).
\[l\imath mx->0{(arc\tan{(\frac{{x}^{2}-\sqrt{3}}{(x-1)({x}^{2}+x+1)})})}^{\frac{x}{\sin{2x}}}\]
Regroup terms.
\[l\imath mx->0{((\tan{(\frac{{x}^{2}-\sqrt{3}}{(x-1)({x}^{2}+x+1)})})arc)}^{\frac{x}{\sin{2x}}}\]
Use Multiplication Distributive Property : \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[l\imath mx->0\tan^{\frac{x}{\sin{2x}}}(\frac{{x}^{2}-\sqrt{3}}{(x-1)({x}^{2}+x+1)}){a}^{\frac{x}{\sin{2x}}}{r}^{\frac{x}{\sin{2x}}}{c}^{\frac{x}{\sin{2x}}}\]
Simplify \(0\tan^{\frac{x}{\sin{2x}}}(\frac{{x}^{2}-\sqrt{3}}{(x-1)({x}^{2}+x+1)}){a}^{\frac{x}{\sin{2x}}}{r}^{\frac{x}{\sin{2x}}}{c}^{\frac{x}{\sin{2x}}}\) to \(0\).
\[l\imath mx->0\]
Regroup terms.
\[-+l\imath mx>0\]
Divide both sides by \(\imath \).
\[-+lmx>0\]
Divide both sides by \(m\).
\[-+lx>0\]
Divide both sides by \(x\).
\[-+l>0\]
Multiply both sides by \(-1\).
\[l<0\]
l<0
