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

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

Commented bymalwan last updated on 23/Sep/20

can we solve it with lhopital??

Answered by john santu last updated on 23/Sep/20

lim_(x→∞)  (√(x^2 +2x)) ((√(x^2 +1))−(√(x^2 +4)))=  lim_(x→∞)  (√(x^2 +2x)) (((−3)/( (√(x^2 +1))+(√(x^2 +4))))) =  −3 ×lim_(x→∞)  ((x (√(1+(2/x))))/(x ((√(1+(1/x)))+(√(1+(4/x)))))) =  −(3/2).

Answered by bemath last updated on 23/Sep/20

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

lim_(x→∞) x^2 (√((1+(2/x))(1+(1/x^2 )))) −x^2 (√((1+(2/x))(1+(4/x^2 ))))  lim_(x→∞) x^2 ((1+(1/x))(1+(1/(2x^2 )))−(1+(1/x))(1+(2/x^2 )))  lim_(x→∞) x^2 ((1+(1/x))((1/(2x^2 ))−(2/x^2 )))  lim_(x→∞) x^2 ((1/(2x^2 ))+(1/(2x^3 ))−(2/x^2 )−(2/x^3 ))=((1/2)−2)=−(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)