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Question Number 159744 by 0731619 last updated on 20/Nov/21

Answered by gsk2684 last updated on 20/Nov/21

$$\frac{0}{0}form,apply{L}^{\prime }hospitalrule$$$$\underset{x\to 2}{\lim }\frac{-\left(\cos \frac{\pi }{x}\right)(-\frac{\pi }{x^2})}{\left(\sin \pi x\right)\pi }=\underset{x\to 2}{\frac{1}{4}\lim \frac{\left(\cos \frac{\pi }{x}\right)}{\left(\sin \pi x\right)}}$$$$againapplythesamerule$$$$\frac{1}{4}\underset{x\to 2}{\lim }\frac{(-\sin \frac{\pi }{x})(-\frac{\pi }{x^2})}{\left(\cos \pi x\right)\pi }=\frac{1}{4}\frac{(-\sin \frac{\pi }{2})(-\frac{\pi }{2^2})}{\left(\cos 2\pi \right)\pi }$$$$=\frac{1}{4}\frac{(-1)(-\frac{\pi }{4})}{\left(1\right)\pi }=\frac{1}{16}$$

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