[tex] \lim_{x \to 0} \frac{1-\cos{x}}{x\sin{x}}
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\\ \lim_{x \to 0} \frac{1-\cos{x}}{x} \frac{1}{\sin{x}}
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\\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \frac{x}{\sin{x}}
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\\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \frac{1}{\frac{\sin{x}}{x}}
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\\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \lim_{x \to 0} \frac{1}{\frac{\sin{x}}{x}}
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\\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \frac{1}{\lim_{x \to 0}\frac{\sin{x}}{x}}
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\\ \frac{1}{2} \times \frac{1}{1} = \frac{1}{2} [/tex]
