calculate-lim-n-1-e-nx-2-x-2-sin-pi-n-

Question Number 40759 by math khazana by abdo last updated on 27/Jul/18

c a l c u l a t e {l i m}_{n \to + \infty} \frac{1 - e^{- {n x}^{2}}}{x^{2} s i n (\frac{π}{n})}

Commented by math khazana by abdo last updated on 27/Jul/18

t h e Q i s f i n d {l i m}_{x \to 0} \frac{1 - e^{- {n x}^{2}}}{x^{2} s i n (\frac{π x}{n})}

Commented by tanmay.chaudhury50@gmail.com last updated on 27/Jul/18

l i \underset{x \to 0}{m} \frac{e^{{n x}^{2}} - 1}{{n x}^{2}} \times \frac{n}{e^{{n x}^{2}}} \times \frac{1}{{\frac{s i n (\frac{Π x}{n})}{\frac{Π x}{n}}}} \times \frac{1}{\frac{Π x}{n}}

= 1 \times \frac{n}{1} \times \frac{1}{1} \times \frac{n}{0} = \infty

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Jul/18

l i \underset{n \to \infty}{m} \frac{1 - \frac{1}{e^{{n x}^{2}}}}{x^{2} s i n (\frac{Π}{n})}

w h e n n \to \infty \frac{1}{e^{{n x}^{2}}} \to 0 a n d f o r a n y v a l u e o f

n t h e v a l u e o f s i n (\frac{Π}{n}) l i e s b e t w e e n \pm 1

\frac{1}{x^{2}} > l i \underset{n \to \infty}{m} \frac{1 - e^{- {n x}^{2}}}{x^{2} s i n (\frac{Π}{n})} > - \frac{1}{x^{2}}

\underset{t \to 0}{li}