lim_(x→∞) (√((x^2 +2x)(x^2 +1))) −(√((x^2 +2x)(x^2 +4))) ?


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Question Number 115122 by bobhans last updated on 23/Sep/20

$\lim_{x \to \infty} \sqrt{(x^{2} + 2 x) (x^{2} + 1)} - \sqrt{(x^{2} + 2 x) (x^{2} + 4)} ?$
Commented by malwan last updated on 23/Sep/20

$c a n w e s o l v e i t w i t h l h o p i t a l ? ?$
	Answered by john santu last updated on 23/Sep/20

	$\lim_{x \to \infty} \sqrt{x^{2} + 2 x} (\sqrt{x^{2} + 1} - \sqrt{x^{2} + 4}) =$ $\lim_{x \to \infty} \sqrt{x^{2} + 2 x} (\frac{- 3}{\sqrt{x^{2} + 1} + \sqrt{x^{2} + 4}}) =$ $- 3 \times \lim_{x \to \infty} \frac{x \sqrt{1 + \frac{2}{x}}}{x (\sqrt{1 + \frac{1}{x}} + \sqrt{1 + \frac{4}{x}})} =$ $- \frac{3}{2} .$
	Answered by bemath last updated on 23/Sep/20

	Answered by Dwaipayan Shikari last updated on 23/Sep/20

	${\underset{x \to \infty}{\lim x}}^{2} \sqrt{(1 + \frac{2}{x}) (1 + \frac{1}{x^{2}})} - x^{2} \sqrt{(1 + \frac{2}{x}) (1 + \frac{4}{x^{2}})}$ ${\underset{x \to \infty}{\lim x}}^{2} ((1 + \frac{1}{x}) (1 + \frac{1}{{2 x}^{2}}) - (1 + \frac{1}{x}) (1 + \frac{2}{x^{2}}))$ ${\underset{x \to \infty}{\lim x}}^{2} ((1 + \frac{1}{x}) (\frac{1}{{2 x}^{2}} - \frac{2}{x^{2}}))$ ${\underset{x \to \infty}{\lim x}}^{2} (\frac{1}{{2 x}^{2}} + \frac{1}{{2 x}^{3}} - \frac{2}{x^{2}} - \frac{2}{x^{3}}) = (\frac{1}{2} - 2) = - \frac{3}{2}$
	Answered by Bird last updated on 24/Sep/20
	$let f(x) =(√((x^2 +2x)(x^2 +1)))−(√((x^2 +2x)(x^2 +4))) f(x)=x^2 (√((1+(2/x)))).(√(1+(1/x^2 )))−x^2 (√(1+(2/x))).(√(1+(4/x^2 ))) f(x)∼x^2 (1+(1/x))(1+(1/(2x^2 ))) −x^2 (1+(1/x))(1+(2/x^2 ))(x→∞) ⇒ f(x)∼(1+(1/x)){x^2 +(1/2)−x^2 −2} f(x)∼(1+(1/x))(−(3/2)) ⇒ lim_(x→∞) f(x) =−(3/2)$
	$l e t f (x) = \sqrt{(x^{2} + 2 x) (x^{2} + 1)} - \sqrt{(x^{2} + 2 x) (x^{2} + 4)}$ $f (x) = x^{2} \sqrt{(1 + \frac{2}{x})} . \sqrt{1 + \frac{1}{x^{2}}} - x^{2} \sqrt{1 + \frac{2}{x}} . \sqrt{1 + \frac{4}{x^{2}}}$ $f (x) \sim x^{2} (1 + \frac{1}{x}) (1 + \frac{1}{2 x^{2}})$ $- x^{2} (1 + \frac{1}{x}) (1 + \frac{2}{x^{2}}) (x \to \infty) \Rightarrow$ $f (x) \sim (1 + \frac{1}{x}) {x^{2} + \frac{1}{2} - x^{2} - 2}$ $f (x) \sim (1 + \frac{1}{x}) (- \frac{3}{2}) \Rightarrow$ ${l i m}_{x \to \infty} f (x) = - \frac{3}{2}$
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