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

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Consider $f(x)=\lim\limits_{x \to 0} \csc x-\cot x=\lim\limits_{x \to 0} \dfrac{1}{\sin x}-\lim\limits_{x \to 0} \dfrac{\cos x}{\sin x}$ This can be further simplified as: $\lim\limits_{x \to 0} \dfrac{1}{\sin x}-\lim\limits_{x \to 0} \dfrac{\cos x}{\sin x}=\lim\limits_{x \to 0} \dfrac{(1-\cos x )\sin x}{(1-cos x)(1+\cos x)}$ Then $\lim\limits_{x \to 0}\dfrac{\sin x}{1+\cos x}=0$