Evaluate: $\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta – \sin \theta}{\theta – \dfrac {\pi}{4}}$.
My Attempt:
\begin{align}
\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta – \sin \theta }{\theta – \dfrac {\pi}{4}}
&=\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta – \cos \dfrac {\pi}{4} + \sin \dfrac {\pi}{4} – \sin \theta}{\theta – \dfrac {\pi}{4}}
\\
&=\lim_{\theta \to \frac {\pi}{4}} \dfrac {2\sin \dfrac {\pi-4\theta }{8}\cos \dfrac {\pi+4\theta}{8} – 2\sin \dfrac {4\theta + \pi}{8}\sin \dfrac {4\theta -\pi}{8}}{\theta – \dfrac {\pi}{4}}.
\end{align}
How do I proceed?
Best Answer
Add and subtract $\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right)$ in the numerator to get:
$$\lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\cos \theta - \sin \theta}{\theta - \dfrac {\pi}{4}} = \lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\cos \theta - \cos\left(\frac{\pi}{4}\right) - \left(\sin \theta - \sin\left(\frac{\pi}{4}\right)\right)}{\theta - \dfrac {\pi}{4}} $$ $$= \lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\cos \theta - \cos\left(\frac{\pi}{4}\right)}{\theta - \dfrac {\pi}{4}} - \lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\sin \theta - \sin\left(\frac{\pi}{4}\right)}{\theta - \dfrac {\pi}{4}} = -\sin\left(\dfrac {\pi}{4}\right) - \cos\left(\dfrac {\pi}{4}\right) = -\sqrt{2}$$
where we used the definition for derivatives of sine and cosine.