((JS)/(■★★■)) lim_(x→3) ((5−2x+(√(x−2)))/(x−4+(√(x−2)))) ?


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Question Number 109861 by john santu last updated on 26/Aug/20

$\frac{J S}{◼ ★ ★ ◼}$ $\lim_{x \to 3} \frac{5 - 2 x + \sqrt{x - 2}}{x - 4 + \sqrt{x - 2}} ?$
	Answered by bemath last updated on 26/Aug/20

	Commented by john santu last updated on 26/Aug/20

	$b e a u t i f u l l$
	Answered by Her_Majesty last updated on 26/Aug/20

	$\frac{5 - 2 x + \sqrt{x - 2}}{x - 4 + \sqrt{x - 2}} = \frac{6 - 2 x + 3 \sqrt{x - 2}}{x - 6} \Rightarrow l i m = - 1$
	Commented by bemath last updated on 26/Aug/20

	$w h a t y o u r m e t h o d m i s s ?$
	Commented by Her_Majesty last updated on 26/Aug/20

	$\frac{5 - 2 x + \sqrt{x - 2}}{x - 4 + \sqrt{x - 2}} \times \frac{x - 4 - \sqrt{x - 2}}{x - 4 - \sqrt{x - 2}} =$ $= \frac{(x - 3) (6 - 2 x + 3 \sqrt{x - 2})}{(x - 3) (x - 6)} = \frac{6 - 2 x + 3 \sqrt{x - 2}}{x - 6}$
	Commented by bemath last updated on 26/Aug/20

	$c o o l l . . . . n i c e$
	Answered by 1549442205PVT last updated on 26/Aug/20

	$\lim_{x \to 3} \frac{5 - 2 x + \sqrt{x - 2}}{x - 4 + \sqrt{x - 2}} \underset{L^{'} Hopital}{=} \lim_{x \to 3} \frac{- 2 + \frac{1}{2 \sqrt{x - 2}}}{1 + \frac{1}{2 \sqrt{x - 2}}}$ $= \frac{- 2 + \frac{1}{2}}{1 + \frac{1}{2}} = \frac{- 3 / 2}{3 / 2} = - 1$
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