lim_(x→∞) [ x (√((x−1)/(9x+2))) −(x/3) ]?


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Question Number 123349 by bemath last updated on 25/Nov/20

$\lim_{x \to \infty} [x \sqrt{\frac{x - 1}{9 x + 2}} - \frac{x}{3}] ?$
	Answered by liberty last updated on 25/Nov/20

	$\lim_{x \to \infty} x [\sqrt{\frac{x - 1}{9 x + 2}} - \frac{1}{3}] =$ $\lim_{x \to \infty} \frac{x (3 \sqrt{x - 1} - \sqrt{9 x + 2})}{3 \sqrt{9 x + 2}} =$ $\lim_{x \to \infty} \frac{x (\sqrt{9 x - 9} - \sqrt{9 x + 2})}{\sqrt{9 x + 2}} =$ $\lim_{x \to \infty} \frac{x (9 x - 9 - 9 x - 2)}{3 \sqrt{9 x + 2} (\sqrt{9 x - 9} + \sqrt{9 x + 2})} =$ $\lim_{x \to \infty} \frac{- 11 x}{3 x \sqrt{9 + \frac{2}{x}} (\sqrt{9 - \frac{9}{x}} + \sqrt{9 + \frac{2}{x}})} =$ $\frac{- 11}{9 (3 + 3)} = - \frac{11}{54}$
	Commented by bemath last updated on 25/Nov/20

	$t y p o i t s h o u l d b e - \frac{11}{54}$
	Commented by liberty last updated on 25/Nov/20

	$y e s .$
	Answered by Bird last updated on 25/Nov/20
	$let f(x)=x(√((x−1)/(9x+2)))−(x/3) ⇒f(x)=x(√((1/9)×((x−1)/(x+(2/9)))))−(x/3) =(x/3){(√((x+(2/9)−1−(2/9))/(x+(2/9))))−1} =(x/3){(√(1−((11)/(9x+2))))−1} ∼(x/3){1−((11)/(2(9x+2)))−1} =−((11x)/(6(8x+2))) ⇒ lim_(x→+∞) f(x)=−((11)/(48))$
	$l e t f (x) = x \sqrt{\frac{x - 1}{9 x + 2}} - \frac{x}{3}$ $\Rightarrow f (x) = x \sqrt{\frac{1}{9} \times \frac{x - 1}{x + \frac{2}{9}}} - \frac{x}{3}$ $= \frac{x}{3} {\sqrt{\frac{x + \frac{2}{9} - 1 - \frac{2}{9}}{x + \frac{2}{9}}} - 1}$ $= \frac{x}{3} {\sqrt{1 - \frac{11}{9 x + 2}} - 1}$ $\sim \frac{x}{3} {1 - \frac{11}{2 (9 x + 2)} - 1}$ $= - \frac{11 x}{6 (8 x + 2)} \Rightarrow$ ${l i m}_{x \to + \infty} f (x) = - \frac{11}{48}$
	Commented by Bird last updated on 25/Nov/20

	$e r r o r o f t y p o f (x) \sim - \frac{11 x}{6 (9 x + 2)}$ $\Rightarrow {l i m}_{x \to + \infty} f (x) = - \frac{11}{6.9} = - \frac{11}{54}$
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