Question

Evaluate on lim x->0 (1-cosx) / x sinx

Answer 1

$\lim_{x \to 0} \frac{1-\cos{x}}{x\sin{x}} \\ \\ \lim_{x \to 0} \frac{1-\cos{x}}{x} \frac{1}{\sin{x}} \\ \\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \frac{x}{\sin{x}} \\ \\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \frac{1}{\frac{\sin{x}}{x}} \\ \\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \lim_{x \to 0} \frac{1}{\frac{\sin{x}}{x}} \\ \\\lim_{x \to 0} \frac{1-\cos{x}}{x^2} \frac{1}{\lim_{x \to 0}\frac{\sin{x}}{x}} \\ \\ \frac{1}{2} \times \frac{1}{1} = \frac{1}{2}$