Single Variable Calculus, Chapter 7, 7.8, Section 7.8, Problem 44

Determine the $\displaystyle \lim_{x \to \pi/4} (1- \tan x) \sec x$. Use L'Hospital's Rule where appropriate. Use some Elementary method if posible. If L'Hospitals Rule doesn't apply. Explain why.


$
\begin{equation}
\begin{aligned}
\lim_{x \to \pi/4} (1- \tan x) \sec x &= \lim_{x \to \pi/4} \left( 1 - \frac{\sin x}{\cos x} \right) \frac{1}{\cos x}\\
\\
&= \lim_{x \to \pi/4} \left( \frac{1}{\cos x} - \frac{\sin x}{\cos^2 x} \right)\\
\\
&= \frac{1}{\cos \frac{\pi}{4}} - \frac{\sin \frac{\pi}{4}}{\left( \cos \frac{\pi}{4} \right)^2}\\
\\
&= \frac{1}{\frac{\sqrt{2}}{2}} - \frac{\frac{\sqrt{2}}{2}}{\left( \frac{\sqrt{2}}{2} \right)^2}\\
\\
&= \frac{2}{\sqrt{2}} - \frac{2}{\sqrt{2}} = 0
\end{aligned}
\end{equation}
$