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Question
If \[\lim_{x \to 0} kx cosec x = \lim_{x \to 0} x cosec kx,\]
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Solution
\[\lim_{x \to 0} kx . cosec x = \lim_{x \to 0} x cosec kx\]
\[ \Rightarrow \lim_{x \to 0} \left[ \frac{kx}{\sin x} \right] = \lim_{x \to 0} \left[ \frac{x}{\sin \left( kx \right)} \right]\]
\[ \Rightarrow k \lim_{x \to 0} \left[ \frac{x}{\sin x} \right] = \lim_{x \to 0} \left[ \frac{kx}{\sin \left( kx \right)} \right] \times \frac{1}{k}\]
\[ \Rightarrow k = \frac{1}{k}\]
\[ \Rightarrow k^2 = 1\]
\[ \Rightarrow k = \pm 1\]
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