calculate lim_(x→1) ((1+cos(πx))/(x^2 − sin(((πx)/2))))


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Question Number 39635 by math khazana by abdo last updated on 09/Jul/18

$c a l c u l a t e {l i m}_{x \to 1} \frac{1 + c o s (π x)}{x^{2} - s i n (\frac{π x}{2})}$
Commented by abdo mathsup 649 cc last updated on 09/Jul/18

$c h a n g e m e n t x - 1 = t g i v e t \to 0 w h e n x \to 1 a n d$ $\frac{1 + c o s (π x)}{x^{2} - s i n (\frac{π x}{2})} = \frac{1 + c o s (π (1 + t))}{{(1 + t^{})}^{2} - s i n (\frac{π (1 + t)}{2})}$ $= \frac{1 - c o s t}{1 + t^{2} + 2 t - c o s (\frac{π t}{2})} = A (t) b u t$ $1 - c o s t \sim \frac{t^{2}}{2} a n d 1 - c o s (\frac{π t}{2}) \sim \frac{π^{2} t^{2}}{8} (t \to 0) \Rightarrow$ $A (t) \sim \frac{\frac{t^{2}}{2}}{t^{2} + 2 t + \frac{π^{2} t^{2}}{8}} = \frac{t}{2 (t + 2 + \frac{π^{2} t}{8})}_{t \to 0} \to 0$ $\Rightarrow {l i m}_{x \to 1} \frac{1 + c o s (π x)}{x^{2} - s i n (\frac{π x}{2})} = 0$
	Answered by tanmay.chaudhury50@gmail.com last updated on 09/Jul/18
	$t=1−x lim_(t→0) ((1+cos{Π(1−t)})/((1−t)^2 −sin{((Π(1−t))/2)})) lim_(t→0) ((1−cos(Πt))/((1−t)^2 −cos((Πt)/2))) lim_(t→0) ((2sin^2 ((Πt)/2))/(1−2t+t^2 −cos((Πt)/2))) lim_(t→0) ((2sin^2 ((Πt)/2))/(2sin^2 ((Πt)/2)+t^2 −2t)) lim_(t→0) ((2×((sin(((Πt)/2))×sin(((Πt)/2)))/(((Πt)/2)×((Πt)/2)))×(Π^2 /4))/(2×((sin(((Πt)/2))×sin(((Πt)/2)))/(((Πt)/2)×((Πt)/2)))×(Π^2 /4)+1−(2/t))) lim_(t→0) ((Π^2 /2)/((Π^2 /2)+1−∞))=0 pls check...$
	$t = 1 - x$ $\lim_{t \to 0} \frac{1 + c o s {Π (1 - t)}}{{(1 - t)}^{2} - s i n {\frac{Π (1 - t)}{2}}}$ $\lim_{t \to 0} \frac{1 - c o s (Π t)}{{(1 - t)}^{2} - c o s \frac{Π t}{2}}$ $\lim_{t \to 0} \frac{2 {s i n}^{2} \frac{Π t}{2}}{1 - 2 t + t^{2} - c o s \frac{Π t}{2}}$ $\lim_{t \to 0} \frac{2 {s i n}^{2} \frac{Π t}{2}}{2 {s i n}^{2} \frac{Π t}{2} + t^{2} - 2 t}$ $\lim_{t \to 0} \frac{2 \times \frac{s i n (\frac{Π t}{2}) \times s i n (\frac{Π t}{2})}{\frac{Π t}{2} \times \frac{Π t}{2}} \times \frac{\prod^{2}}{4}}{2 \times \frac{s i n (\frac{Π t}{2}) \times s i n (\frac{Π t}{2})}{\frac{Π t}{2} \times \frac{Π t}{2}} \times \frac{\prod^{2}}{4} + 1 - \frac{2}{t}}$ $\lim_{t \to 0} \frac{\frac{\prod^{2}}{2}}{\frac{\prod^{2}}{2} + 1 - \infty} = 0$ $p l s c h e c k . . .$
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