lim_(x→4) ( ((asin (x−4) + cos πx −1)/(x−4)) )^((x−2)/(x−3)) = 4 Find ′a′ ?


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Question Number 35939 by rahul 19 last updated on 26/May/18

$\lim_{x \to 4} {(\frac{a \sin (x - 4) + \cos π x - 1}{x - 4})}^{\frac{x - 2}{x - 3}} = 4$ $F i n d^{'} a^{'} ?$
	Answered by tanmay.chaudhury50@gmail.com last updated on 26/May/18
	$t=x−4 ((lim)/(t→0))×{((asint+cosΠ(t+4)−1)/t)}^((t+2)/(t+1)) =((lim)/(t→0))×{((asint+cosΠt−1)/t)}^((t+2)/(t+1)) =((lim)/(t→0))×{((asint)/t)−((2sin^2 (((Πt)/2)))/t)}^((t+2)/(t+1)) =((lim)/(t→0))×{((asint)/t)−2((sin(((Πt)/2)))/((((Πt)/2))))×(Π/2)×sin((Πt)/2)}^((t+2)/(t+1)) =a^2 so a^2 =4 a±2$
	$t = x - 4$ $\frac{l i m}{t \to 0} \times {\frac{a s i n t + c o s Π (t + 4) - 1}{t}}^{\frac{t + 2}{t + 1}}$ $= \frac{l i m}{t \to 0} \times {\frac{a s i n t + c o s Π t - 1}{t}}^{\frac{t + 2}{t + 1}}$ $= \frac{l i m}{t \to 0} \times {\frac{a s i n t}{t} - \frac{2 {s i n}^{2} (\frac{Π t}{2})}{t}}^{\frac{t + 2}{t + 1}}$ $= \frac{l i m}{t \to 0} \times {\frac{a s i n t}{t} - 2 \frac{s i n (\frac{Π t}{2})}{(\frac{Π t}{2})} \times \frac{Π}{2} \times s i n \frac{Π t}{2}}^{\frac{t + 2}{t + 1}}$ $= a^{2}$ $s o a^{2} = 4 a \pm 2$
	Commented by rahul 19 last updated on 26/May/18
	Is a=-2 acceptable ?
	Commented by rahul 19 last updated on 26/May/18

	$T h a n k Y o u s i r .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 26/May/18

	$s o f a r n o p r o b l e m c o n s i d e r i n g a = - 2 i n t h e$ $q u e s t i o n . . .$
	Commented by Joel579 last updated on 26/May/18

	$Sir Tanmay, you can tap the^{'} \sin^{'} button next to$ $the^{'} {SEL}^{'} button, and then press^{'} \lim_{x \to 0}^{'} button$ ${It}^{'} s just tips so you {won}^{'} t type limit manually$
	Commented by tanmay.chaudhury50@gmail.com last updated on 26/May/18

	$t h a n k y o u s i r . .$
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