Question

$\underset{x\to 0}{\lim }\frac{1-\cos 2x}{3{\tan }^{2}x}$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{1-\cos \left(2x\right)}{3{\tan }^{2}x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2{\sin }^{2}x}{3{\tan }^{2}x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{2}{3}\times \frac{{\sin }^{2}x}{{\sin }^{2}x}\times {\cos }^{2}x\right] \\ =\frac{2}{3}{\cos }^{2}0 \\ =\frac{2}{3} \end{gathered}$$