## University Calculus: Early Transcendentals (3rd Edition)

$$\lim_{x\to0}6x^2(\cot x)(\csc 2x)=3$$
$$A=\lim_{x\to0}6x^2(\cot x)(\csc 2x)$$ $$A=\lim_{x\to0}6x^2\Big(\frac{\cos x}{\sin x}\Big)\Big(\frac{1}{\sin2x}\Big)$$ $$A=\lim_{x\to0}6x^2\Big(\frac{\cos x}{\sin x}\Big)\Big(\frac{1}{2\sin x\cos x}\Big)$$ $$A=\lim_{x\to0}6x^2\Big(\frac{1}{2\sin^2x}\Big)=\lim_{x\to0}\frac{3x^2}{\sin^2x}$$ $$A=3\lim_{x\to0}\frac{x^2}{\sin^2x}$$ $$A=3\lim_{x\to0}\Big(\frac{\sin x}{x}\Big)^{-2}=3\Big(\lim_{x\to0}\frac{\sin x}{x}\Big)^{-2}$$ Apply Theorem 7 with $\theta=x$ here: $$A=3\times(1)^{-2}=3\times1=3$$