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Ta có:
\mathop {\lim }\limits_{x \to 0}\left(\cot x-\frac{1}{\sin x}\right)
=\mathop {\lim }\limits_{x \to 0}\frac{\cos x-1}{\sin x}
=\mathop {\lim }\limits_{x \to 0}\frac{-2\sin^2\frac{x}{2}}{2\sin\frac{x}{2}\cos\frac{x}{2}}
=\mathop {\lim }\limits_{x \to 0}\frac{\sin\frac{x}{2}}{\cos\frac{x}{2}}=0
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