Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 88

Express the limit as a derivative and evaluate $\displaystyle \lim_{\theta \to \pi/3} \frac{\cos \theta - 0.5}{\displaystyle \theta - \frac{\pi}{3} }$

Based from the definition,

$\displaystyle f'(a) = \lim_{\theta \to a} \frac{f(\theta) - f(a)}{\theta - a}$

If we let $f(\theta) = \cos \theta$, then $\displaystyle f\left( \frac{\pi}{3} \right) = \cos \left( \frac{\pi}{3} \right) = 0.5 $/p>


$\displaystyle \lim_{\theta \to \pi/3} \frac{\cos \theta - 0.5}{\displaystyle \theta - \frac{\pi}{3}} = \lim_{\theta \to a/3} \frac{\displaystyle f(\theta) - f\left( \frac{\pi}{3} \right)}{\theta - \frac{\pi}{3}}$

Therefore, we have


$
\begin{equation}
\begin{aligned}

f(\theta) =& \cos \theta
\\
\\
f'(\theta) =& - \sin \theta
\\
\\
& \text{So,}
\\
\\
f'\left( \frac{\pi}{3} \right) =& - \sin \left( \frac{\pi}{3} \right)
\\
\\
f'\left( \frac{\pi}{3} \right) =& \frac{- \sqrt{3}}{2 }

\end{aligned}
\end{equation}
$