Chapter 2: Limits and Continuity - Section 2.2 - Limit of a Function and Limit Laws - Exercises 2.2 - Page 57: 46

$\sqrt{3}$

Note that, $\lim\limits_{x \to a}\sin x=\sin a$ and $\lim\limits_{x \to a}\cos x=\cos a$ for all real numbers $a$. Using this along with our limit laws, we get $\lim\limits_{x \to \frac{\pi}{3}\tan x}=\lim\limits_{x \to \frac{\pi}{3}}\dfrac{\sin x}{\cos x}=\dfrac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}}=\dfrac{(\frac{\sqrt{3}}{2})}{(\frac{1}{2})}=\sqrt{3}.$