Chapter 2: Limits and Continuity - Section 2.2 - Limit of a Function and Limit Laws - Exercises 2.2 - Page 57: 45

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Note that, $\lim\limits_{x \to a}\cos x=\cos a$ for all real numbers $a$. Using this along with our limit laws, we get $\lim\limits_{x \to 0}\sec x=\lim\limits_{x \to 0}\dfrac{1}{\cos x}=\dfrac{1}{\cos 0}=\dfrac{1}{1}=1.$