The limand is continuous at x = π/6, so by direct substitution
[tex]\displaystyle \lim_{x\to\pi/6} \frac{x^2}{\tan(x)} = \frac{\left(\frac\pi6\right)^2}{\tan\left(\frac\pi6\right)} = \frac{\pi^2}{36\times\frac1{\sqrt3}} = \boxed{\frac{\pi^2}{12\sqrt3}}[/tex]
