1. How can I prove that if a complex valued function of complex variable, f(z), is analytic in a region where f(z) isn't zero for all z in that region, 1/f(z) is also analytic in that region.

2.Why does $\lim_{z \to a} \left (z-a)f(z) \right = 0$ imply the existence of $\lim_{z \to a} f(z)$? Here f(z) is a complex valued function of complex variable