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Question
\[\lim_{x \to 0} \frac{\tan mx}{\tan nx}\]
Solution
\[\lim_{x \to 0} \left[ \frac{\tan mx}{\tan nx} \right]\]
\[\Rightarrow \lim_{x \to 0} \left( \frac{\tan mx}{mx} \times \frac{mx}{nx} \times \frac{nx}{\tan nx} \right)\]
\[ \Rightarrow \lim_{x \to 0} \left( \frac{\tan mx}{mx} \times \frac{mx}{\frac{\tan nx}{nx}} \times nx \right)\]
\[ \Rightarrow \frac{m}{n} \left[ \because \lim_{x \to 0} \frac{\tan x}{x} = 1 \right]\]
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