calculate-lim-x-1-1-cos-pix-x-2-sin-pix-2-

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 \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

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