1. ## Triogo-limit

Calculate : $\150dpi \lim_{x\rightarrow 1}\frac{sin\left ( \frac{\pi }{2x} \right )-1}{x^{3}-1}$

2. Originally Posted by dhiab
Calculate : $\150dpi \lim_{x\rightarrow 1}\frac{sin\left ( \frac{\pi }{2x} \right )-1}{x^{3}-1}$
Since this $\to \frac{0}{0}$ you can use L'Hospital's Rule...

$\lim_{x \to 1}\frac{\sin{\left(\frac{\pi}{2x}\right)} - 1}{x^3 - 1} = \lim_{x \to 1}\frac{-\frac{\pi}{2x^2}\cos{\left(\frac{\pi}{2x}\right)}} {3x^2}$

$= \frac{-\frac{\pi}{2(1)^2}\cos{\left[\frac{\pi}{2(1)}\right]}}{3(1)^2}$

$= 0$.