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Question Number 69809 by RAKESH MANDA last updated on 28/Sep/19

$$\underset{x\to \mathrm{\infty }}{\lim }x\sin \frac{\pi }{x}$$

Commented by mr W last updated on 28/Sep/19

$$\underset{x\to \mathrm{\infty }}{\lim }x\sin \frac{\pi }{x}$$$$=\pi \underset{x\to \mathrm{\infty }}{\lim }\frac{\sin \frac{\pi }{x}}{\frac{\pi }{x}}$$$$=\pi \underset{t\to 0}{\lim }\frac{\sin t}{t}$$$$=\pi \times 1$$$$=\pi$$

Commented by peter frank last updated on 28/Sep/19

$$thankyou$$

Answered by $@ty@m123 last updated on 28/Sep/19

$$Letx=\frac{1}{y}$$$$sothatasx\to \mathrm{\infty },y\to 0$$$$\therefore \underset{x\to \mathrm{\infty }}{\lim }x\sin \frac{\pi }{x}$$$$=\underset{y\to 0}{\lim }\frac{\sin \pi y}{y}$$$$=\underset{y\to 0}{\lim }\frac{\sin \pi y}{\pi y}\times \pi$$$$=1\times \pi$$$$=\pi$$

Answered by malwaan last updated on 28/Sep/19

$$\underset{\mathit{x}\to \mathrm{\infty }}{\mathbf{lim}}\frac{\mathbf{sin}\frac{\mathit{\pi }}{\mathit{x}}}{\frac{\mathit{\pi }}{\mathit{x}}}\times \mathit{\pi }$$$$=\mathit{\pi }\times \underset{\mathit{t}\to 0}{\mathbf{lim}}\frac{\mathbf{sin}\mathit{t}}{\mathit{t}}=\mathit{\pi }$$

Commented by malwaan last updated on 28/Sep/19

$$Ingeneral$$$$\underset{\mathit{x}\to \mathrm{\infty }}{\mathbf{lim}}\mathit{a}\mathit{x}\mathbf{sin}\frac{\mathit{b}}{\mathit{c}\mathit{x}}=\frac{\mathit{a}\mathit{b}}{\mathit{c}}$$

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