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Question
\[\lim_{x \to 0} \frac{x^2 - \tan 2x}{\tan x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{x^2 - \tan 2x}{\tan x} \right]\]
\[\text{ Dividing the numerator and the denominator by } x:\]
\[ \lim_{x \to 0} \left[ \frac{x - \frac{\tan 2x}{x}}{\left( \frac{\tan x}{x} \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{x - \frac{\tan 2x}{2x} \times 2}{\left( \frac{\tan x}{x} \right)} \right]\]
\[ = \frac{0 - 2\left( 1 \right)}{1}\]
\[ = - 2\]
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