Evaluate the following limit : lim(x→π/2) ((π/2 - x) sin x - 2 cos x)/((π/2 - x) + cot x)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to\pi/2}\cfrac{(\frac{\pi}2-\text x)sin\,\text x-2\,cos\,\text x}{(\frac{\pi}2-\text x)+cot\,\text x} \)

lim(x→π/2) ((π/2 - x) sin x - 2 cos x)/((π/2 - x) + cot x)

Answer 1

We have Given, \(\lim\limits_{\text x \to\pi/2}\cfrac{(\frac{\pi}2-\text x)sin\,\text x-2\,cos\,\text x}{(\frac{\pi}2-\text x)+cot\,\text x} \)

Hence, \(\lim\limits_{\text x \to\pi/2}\cfrac{(\frac{\pi}2-\text x)sin\,\text x-2\,cos\,\text x}{(\frac{\pi}2-\text x)+cot\,\text x} \) = \(-\cfrac12\)