Evaluate the following limit$.$

$\underset{0}{lim}\frac{\frac{1}{2sin}-\frac{1}{2}}{sin}$

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A$.$

$\underset{0}{lim}\frac{\frac{1}{2sin}-\frac{1}{2}}{sin}=,$ because $sinis$ continuous for all $$

$($Simplity your answer.$)$

B$.$ The limit does not exist and is neither $$ nor $-.$