Question Number 148654 by mathdanisur last updated on 29/Jul/21

lim_(x→1) (((1+x)/(2+x)))^((1−(√x))/(1−x)) = ?

Answered by ArielVyny last updated on 29/Jul/21

e^(((1−(√x))/(1−x))ln(((1+x)/(2+x)))) posons t=x−1 { ((x→1)),((t→0)) :} e^((((√(t+1))−1)/t)ln(((t+2)/(t+3)))) =_(t→0) e^((((√(t+1))−1)/t)ln((2/3))) ∼e^(ln((2/3))) =(2/3) (((√(t+1))−1)/t)∼_0 (t/t)∼1 conclusion lim_(x→1) (((1+x)/(2+x)))^((1−(√x))/(1−x)) =(2/3)

Commented bymathdanisur last updated on 29/Jul/21

Thank you Ser, sqr(2/3)

Commented byArielVyny last updated on 29/Jul/21

yes you have rigth

Answered by Rasheed.Sindhi last updated on 29/Jul/21

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

Commented bymathdanisur last updated on 29/Jul/21

Thank you Ser

Answered by liberty last updated on 30/Jul/21

T=lim_(x→1) (((2+x−1)/(2+x)))^((1−(√x))/((1+(√x))(1−(√x)))) T= lim_(x→1) (1−(1/(2+x)))^(1/(1+(√x))) T=(1−(1/3))^(1/2) =((2/3))^(1/2) =(√(2/3))=((√6)/3)

Commented bymathdanisur last updated on 30/Jul/21

Thank you Ser

Answered by mathmax by abdo last updated on 31/Jul/21

f(x)=(((x+1)/(x+2)))^((1−(√x))/(1−x)) changement 1−x=t give f(x)=f(1−t)=(((2−t)/(3−t)))^((1−(√(1−t)))/t) ⇒ln(f(1−t)=((1−(√(1−t)))/t)ln(((2−t)/(3−t))) =((1−(√(1−t)))/t){ln(2(1−(t/2)))−ln(3(1−(t/3)))} =((1−(√(1−t)))/t){ln((2/3))+ln(1−(t/2))−ln(1−(t/3))} (t→0) ⇒ln(f(1−t))∼((1−(1−(t/2)))/t){ln((2/3))−(t/2)+(t/3)} ⇒ln(f(1−t))∼(1/2){ln((2/3))−(t/6)}→ln((√(2/3))) ⇒ f(1−t)→(√(2/3)) ⇒lim_(x→1) f(x)=(√(2/3))

Commented bymathdanisur last updated on 01/Aug/21

Thank You Ser