# Show that the function $f(x) = | \sin x+\cos x|$ is continuous at $x =\Large\pi$

Toolbox:
• For continuous at $x = \pi, LHL = RHL = f(\pi)$
$LHL\: \lim\limits_{h \to 0} \: | sin(\pi-h)+cos(\pi-h)|$
$LHL\: \lim\limits_{h \to 0} \: | sin\: h-cos\:h | = 1$
RHL $LHL\: \lim\limits_{h \to 0} \: | sin(\pi+h)+cos(\pi+h)|$
= $\lim\limits_{h \to 0} \: | -sin\:h-cos\: h | = 1$
$f(\pi) = | sin\pi+ cos\pi | = 1$
$\Rightarrow f$ is continuous at $x = \pi$

edited Mar 26, 2013