Discuss the continuity of functionf(x)=xtan((1/x)) at x=0 given tht x≠0 and f(0)=0


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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

	$t a n (y) > y, \forall y ⩾ 0 . . . ?$ $w e h a v e$ $1 + {t g}^{2} (t) ⩾ 1$ $\Rightarrow \int_{0}^{x} (1 + {t g}^{2} (t)) d t ⩾ \int_{0}^{x} 1 . d t, \forall x ⩾ 0$ $\Leftrightarrow t g (x) ⩾ x$ $t g (\frac{1}{x}) ⩾ \frac{1}{x}, \forall x ⩾ 0$ $\Rightarrow \lim_{x \to 0^{+}} . x t g (\frac{1}{x}) ⩾ x . \frac{1}{x} = 1 \Rightarrow \lim_{x \to 0^{+}} x t g (\frac{1}{x}) ⩾ 1$ $n o c o n t i n u i t y$
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