The function $$f\left( x \right) = \frac{{1 - \sin \,x + \cos \,x}}{{1 + \sin \,x + \cos \,x}}$$      is not defined at $$x = \pi .$$  The value of $$f\left( \pi \right)$$  so that $$f\left( x \right)$$  is continuous at $$x = \pi ,$$  is :

Solution :
$$\eqalign{ & \mathop {\lim }\limits_{x \to \pi } \frac{{1 - \sin \,x + \cos \,x}}{{1 + \sin \,x + \cos \,x}} \cr & {\text{Using L'hospital's rules }} \cr & \Rightarrow \mathop {\lim }\limits_{x \to \pi } \frac{{ - \cos \,x - \sin \,x}}{{\cos \,x - \sin \,x}} = \frac{{ - \cos \,\pi - \sin \,\pi }}{{\cos \,\pi - \sin \,\pi }} = \frac{{ - \left( { - 1} \right) - 0}}{{ - 1 - 0}} = - 1 \cr} $$