216 $\\lim_{x \\to \\pi / 2} \\frac{\\cot x}{\\cos x}$

Explanation:

Step1: Rewrite cotangent

Recall that (\cot x=\frac{\cos x}{\sin x}). So (\frac{\cot x}{\cos x}=\frac{\frac{\cos x}{\sin x}}{\cos x}). Simplify the expression: (\frac{\frac{\cos x}{\sin x}}{\cos x}=\frac{1}{\sin x}) (for (x\neq n\pi), (n\in\mathbb{Z})).

Step2: Evaluate the limit

Now we need to find (\lim_{x\rightarrow\frac{\pi}{2}}\frac{1}{\sin x}). Substitute (x = \frac{\pi}{2}) into (\frac{1}{\sin x}). Since (\sin\frac{\pi}{2}=1).

Answer:

(1)