Discuss-the-continuity-of-functionf-x-xtan-1-x-at-x-0-given-tht-x-0-and-f-0-0-

Question Number 125847 by zarminaawan last updated on 14/Dec/20

D i s c u s s t h e c o n t i n u i t y o f f u n c t i o n f (x) = x t a n (\frac{1}{x}) a t x = 0 g i v e n t h t x \neq 0 a n d f (0) = 0

Answered by mindispower last updated on 14/Dec/20

t a n (y) > y, \forall y ⩾ 0 \dots ?

w e h a v e

1 + {t g}^{2} (t) ⩾ 1

\Rightarrow \int_{0}^{x} (1 + {t g}^{2} (t)) d t ⩾ \int_{0}^{x} 1 . d t, \forall x ⩾ 0

\Leftrightarrow t g (x) ⩾ x

t g (\frac{1}{x}) ⩾ \frac{1}{x}, \forall x ⩾ 0

\Rightarrow \lim_{x \to 0^{+}} . x t g (\frac{1}{x}) ⩾ x . \frac{1}{x} = 1 \Rightarrow \lim_{x \to 0^{+}} x t g (\frac{1}{x}) ⩾ 1

n o c o n t i n u i t y