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Question
\[\lim_{x \to 0} \frac{x^3 \cot x}{1 - \cos x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{x^3 \cot x}{1 - \cos x} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{x^3}{\tan x \left( 1 - \cos x \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{x^3}{\tan x \times 2 \sin^2 \left( \frac{x}{2} \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{x}{\tan x} \times \frac{x^2}{2 \sin^2 \left( \frac{x}{2} \right)} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{x}{\tan x} \times \frac{\frac{x^2}{4} \times 4}{2 \times \sin^2 \frac{x}{2}} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{x}{\tan x} \times \left( \frac{\frac{x}{2}}{\sin \frac{x}{2}} \right)^2 \times \frac{4}{2} \right]\]
\[ = 1 \times 1 \times \frac{4}{2}\]
\[ = 2\]
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