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UTS 1 Matematika IIA ITB 2017 (Bagian A nomor 3)

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\begin{align*}\lim_{x\to 0}\left(\frac{1}{\sin x}-\frac{1}{\tan x}\right)&=\lim_{x\to 0}\left(\frac{1}{\sin x}-\frac{\cos x}{\sin x}\right)\\&=\lim_{x\to 0}\frac{1-\cos x}{\sin x}\\&=\lim_{x\to 0}\frac{2\sin^2\frac{1}{2}x}{\sin x}\\&=\lim_{x\to 0}\left(2\cdot\frac{\sin\frac{1}{2}x}{\sin x}\cdot\sin\frac{1}{2}x\right)\\&=2\cdot\frac{1}{2}\cdot 0\\&=\boxed{0}.\end{align*}


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