Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 16

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\frac{1-\cos^2\left(x\right)}{1+\cos\left(x\right)}\right)$

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SOLUTION:

This problem can be solved using a direct substitution of $x=0$ . That is

\begin{align*} \lim\limits_{x\to 0}\left(\frac{1-\cos^2 x }{1+\cos x }\right) & =\frac{1-\cos^2\left(0\right)}{1+\cos\left(0\right)} \\ \\ & =\frac{1-1}{1+1}\\ \\ & = 0 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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