let-f-x-cos-x-cos-1-x-is-f-have-a-limit-at-point-0-

Question Number 40095 by maxmathsup by imad last updated on 15/Jul/18

l e t f (x) = c o s (x) c o s (\frac{1}{x}) i s f h a v e a l i m i t a t p o i n t 0 ?

Answered by math khazana by abdo last updated on 26/Jul/18

l e t x_{n} = \frac{2}{n π} w e h a v e {l i m}_{n \to + \infty} x_{n} = 0 a n d

f (x_{n}) = c o s (\frac{2}{n π}) c o s (\frac{n π}{2}) \Rightarrow

f (x_{2 n}) = c o s (\frac{2}{2 n π}) c o s (\frac{2 n π}{2}) = c o s (\frac{1}{n π}) {(- 1)}^{n}

a n d {(- 1)}^{n} d o n t h a v e a n y l i m i t a n d t h a t p r o v e

t h a t f h a v e n t a n y l i m i t a t p o i n t 0