Lim(X-->pi/2)[cotX / (2x -pi)]

= cot(pi/2) / 2*pi/2 -pi

= 0/0

Indeterminate.

Use the L'Hopital's Rule: Lim(x->a)[f(x)/g(x)] = Lim(x->a)[f'(x)/g'(x)].

= Lim(X->pi/2)[-csc^2(X) / 2

= -csc^2(pi/2) / 2

= -[1/sin(pi/2)]^2 /2

= -[1/1]^2 / 2

= -1/2 ------------answer.