calculate-lim-x-1-sin-pix-1-x-2-without-hospital-rule-

Question Number 79756 by mathmax by abdo last updated on 27/Jan/20

c a l c u l a t e {l i m}_{x \to 1} \frac{s i n (π x)}{1 - x^{2}} w i t h o u t h o s p i t a l r u l e

Commented by john santu last updated on 27/Jan/20

\lim_{x \to 1} (\frac{\sin (π - π x)}{(π - π x)}) \times \lim_{x \to 1} \frac{(π - π x)}{1 - x^{2}}

= 1 \times \lim_{x \to 1} \frac{π (1 - x)}{(1 - x) (1 + x)}

= \frac{π}{2}

Commented by mathmax by abdo last updated on 28/Jan/20

c h a n g e m e n t 1 - x = t g i v e

{l i m}_{x \to 1} \frac{s i n (π x)}{1 - x^{2}} = {l i m}_{t \to 0} \frac{s i n (π (1 - t))}{1 - {(1 - t)}^{2}}

= {l i m}_{t \to 0} \frac{s i n (π t)}{1 - (t^{2} - 2 t + 1)} = {l i m}_{t \to 0} \frac{s i n (π t)}{2 t - t^{2}}

= {l i m}_{t \to 0} \frac{s i n (π t)}{π t} \times \frac{π t}{2 t - t^{2}} = {l i m}_{t \to 0} \frac{s i n (π t)}{π t} \times {l i m}_{t \to 0} \frac{π}{2 - t}

= 1 \times \frac{π}{2} = \frac{π}{2}