lim x arrow (5 pi)/(6)(cos 2 x-csc 3 x)

To evaluate the limit \(\lim_{x \rightarrow \frac{5\pi}{6}} (\cos 2x - \csc 3x)\), we first need to understand the behavior of each trigonometric function as approaches .1. **Evaluate :** As , we have: The cosine of is equivalent to the cosine of (since ), which is: 2. **Evaluate :** As , we have: The cosecant function, , is the reciprocal of the sine function. At , the sine function is: Therefore, \(\csc\left(\frac{5\pi}{2}\right) = \frac{1}{\sin\left(\frac{5\pi}{2}\right)} = 1\).Now, substituting these values into the original expression, we get: Thus, the limit is .