lim_(x→(2/3)) ((⌊3x⌋−3x)/(9x^2 −4)) =?


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Question Number 163642 by cortano1 last updated on 09/Jan/22

$\lim_{x \to \frac{2}{3}} \frac{⌊ 3 x ⌋ - 3 x}{9 x^{2} - 4} = ?$
	Answered by mehdiAz last updated on 09/Jan/22

	$\lim_{x \to \frac{2}{3}} \frac{⌊ 3 x ⌋ - 3 x}{9 x^{2} - 4} = ?$ $i f x \in] - \infty, 0] : \lim_{x \to \frac{2}{3}} \frac{- 3 x - 3 x}{9 x^{2} - 4} = \lim_{x \to \frac{2}{3}} \frac{- 6 x}{9 x^{2} - 4}$ $l e t x^{2} = h \Rightarrow x = - \sqrt{h}$ $w e h a v e x \to \frac{2}{3} \Rightarrow h \to \frac{4}{9}$ $\lim_{h \to \frac{4}{9}} \frac{6 \sqrt{h}}{9 h - 4} = \frac{6 \times \frac{2}{3}}{9 \times \frac{4}{9} - 4} = \frac{4}{4 - 4} = \frac{4}{0^{+}} = + \infty$ $s o \lim_{x \to \frac{2}{3}} \frac{⌊ 3 x ⌋ - 3 x}{9 x^{2} - 4} = + \infty, f o r x \in] - \infty, 0]$ $n o w i f x \in [0, + \infty [, t h e n$ $\lim_{x \to \frac{2}{3}} \frac{3 x - 3 x}{9 x^{2} - 4} = \lim_{x \to \frac{2}{3}} \frac{0}{9 x^{2} - 4} = \lim_{x \to \frac{2}{3}} 0 = 0$
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