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Question
\[\lim_{x \to 0} \frac{1 - \cos 2x}{3 \tan^2 x}\]
Solution
\[\lim_{x \to 0} \left[ \frac{1 - \cos \left( 2x \right)}{3 \tan^2 x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2 \sin^2 x}{3 \tan^2 x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{2}{3} \times \frac{\sin^2 x}{\sin^2 x} \times \cos^2 x \right]\]
\[ = \frac{2}{3} \cos^2 0\]
\[ = \frac{2}{3}\]
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