lim_(x→3) ((cos(((3π)/(2x))))/(x−3))


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Question Number 134378 by mohammad17 last updated on 02/Mar/21

${l i m}_{x \to 3} \frac{c o s (\frac{3 π}{2 x})}{x - 3}$
	Answered by bramlexs22 last updated on 02/Mar/21

	$\lim_{x - 3 \to 0} \frac{\cos (\frac{3 π}{2 x})}{x - 3}$ $let x - 3 = w \Rightarrow \frac{1}{x} = \frac{1}{w + 3}$ $\lim_{w \to 0} \frac{\cos (\frac{3 π}{2 (w + 3)})}{w} =$ $\lim_{w \to 0} \frac{\frac{3 π}{2 {(w + 3)}^{2}} . \sin (\frac{3 π}{2 (w + 3)})}{1} = \frac{π}{6}$
	Commented by mr W last updated on 03/Mar/21

	$\frac{3 π}{2} \times \frac{1}{3^{2}}$
	Commented by bramlexs22 last updated on 02/Mar/21

	$\frac{d}{dw} [\cos (\frac{3 π}{2 (w + 3)})] =$ $\frac{d}{dw} [\cos \frac{3 π}{2} {(w + 3)}^{- 1}] =$ $- \frac{3 π}{2} . {(w + 3)}^{- 2} [- \sin (\frac{3 π}{2 (w + 3)})]$ $\lim_{w \to 0} \frac{3 π}{2 {(w + 3)}^{2}} . \sin (\frac{3 π}{2 (w + 3)}) =$ $\frac{3 π}{18} \times 1 = \frac{π}{6}$
	Answered by mathmax by abdo last updated on 02/Mar/21

	$f (x) = \frac{\cos (\frac{3 π}{2 x})}{x - 3} changement \frac{3 π}{2 x} = \frac{π}{2} + t$ $x \to 3 \Rightarrow t \to 0 and \frac{3 π}{2 x} = \frac{2 t + π}{2} \Rightarrow \frac{3 π}{x} = 2 t + π \Rightarrow \frac{x}{3 π} = \frac{1}{2 t + π} \Rightarrow$ $x = \frac{3 π}{2 t + π} \Rightarrow x - 3 = \frac{3 π}{2 t + π} - 3 = \frac{3 π - 6 t - 3 π}{2 t + π} = \frac{- 6 t}{2 t + π} \Rightarrow$ $f (x) = g (t) = \frac{\cos (\frac{π}{2} + t)}{\frac{- 6 t}{2 t + π}} = \frac{(2 t + π) sint}{6 t} \Rightarrow g (t) \sim \frac{2 t + π}{6} \to \frac{π}{6} \Rightarrow$ $\lim_{x \to 3} f (x) = \frac{π}{6}$
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