calculate-lim-n-x-n-1-cos-pi-x-n-with-x-from-R-and-x-0-

Question Number 37817 by prof Abdo imad last updated on 17/Jun/18

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

f r o m R a n d x \neq 0

Commented by prof Abdo imad last updated on 24/Jun/18

i f ∣ x ∣< 1 {l i m}_{n \to + \infty} x^{n} = 0 c h a n g . x^{n} = \frac{1}{t}

{l i m}_{n \to + \infty} x^{n} (1 - c o s (\frac{π}{x^{n}})) = {l i m}_{t \to + \infty} \frac{1}{t} (1 - c o s (π t))

= {l i m}_{t \to + \infty} \frac{2 {s i n}^{2} (\frac{π t}{2})}{t} = 0 b e c a u s e 0 ⩽ {s i n}^{2} (\frac{π t}{2}) ⩽ 1

i f x > 1 {l i m}_{n \to + \infty} x^{n} = + \infty c h a n g . x^{n} = \frac{1}{t}

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

b u t 1 - c o s (π t) \sim \frac{π^{2} t^{2}}{2} (t \to 0) \Rightarrow

\frac{1 - c o s (π t)}{t} \sim \frac{π^{2} t}{2} \to 0 (t \to 0) s o

{l i m}_{n \to + \infty} x^{n} (1 - c o s (\frac{π}{x^{n}})) = 0 .

Answered by behi83417@gmail.com last updated on 17/Jun/18

\frac{1}{x^{n}} = t, x \to + \infty \Rightarrow t \to 0

L = \lim_{x \to 0} \frac{1 - c o s π t}{t} = \lim_{x \to 0} \frac{\frac{{(π t)}^{2}}{2}}{t} = 0 .

Commented by math khazana by abdo last updated on 18/Jun/18

s i r B e h i b e c a r r e f u l n \to + \infty n o t x \dots