Question Number 149637 by mnjuly1970 last updated on 06/Aug/21

Answered by iloveisrael last updated on 06/Aug/21

lim_(x→∞) ((x^2 (√(1+x^(−1) +x^(−4) )) −x^2 (1+x^(−1) ))/x) = lim_(x→∞) x{(√(1+x^(−1) +x^(−4) ))−(1+x^(−1) )} set x^(−1) = u ∧ u→0 =lim_(u→0) (((√(1+u+u^4 ))−(1+u))/u) =lim_(u→0) (((1+(1/2)(u+u^4 ))−(1+u))/u) =lim_(u→0) ((−(1/2)u+(1/2)u^4 )/u) =lim_(u→0) (−(1/2)+(1/2)u^3 )=−(1/2)

Commented bymnjuly1970 last updated on 06/Aug/21

thx alot..

Answered by EDWIN88 last updated on 06/Aug/21

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

Commented bymnjuly1970 last updated on 06/Aug/21

thank you so much..