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