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Question
\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{x \tan x}{1 - \cos x} \right]\]
\[ {\text{ Dividing the numerator and the denominator by } x}^2:\]
\[ \lim_{x \to 0} \left[ \frac{\frac{x \tan x}{x^2}}{\frac{1 - \cos x}{x^2}} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\frac{\tan x}{x}}{\frac{2 \sin^2 \frac{x}{2}}{\frac{x}{2} \times \frac{x}{2} \times 4}} \right]\]
\[ = \frac{4}{2}\]
\[ = 2\]
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