Question Number 149505 by mathdanisur last updated on 05/Aug/21

lim_(n→∞) (((n + 3)/(n + 1)))^n = ?

Commented byEDWIN88 last updated on 06/Aug/21

lim_(n→∞) (((n+3)/(n+1)))^n = e^(lim_(n→∞) (((n+3)/(n+1))−1)n) =e^(lim_(n→∞) ((2/(n+1)))n) =e^2

Commented bymathdanisur last updated on 06/Aug/21

Thankyou Ser

Answered by Ar Brandon last updated on 06/Aug/21

L=lim_(n→∞) (((n(1+(3/n)))/(n(1+(1/n)))))^n =lim_(n→∞) (1+(3/n))^n /(1+(1/n))^n =lim_(n→∞) e^(nln(1+(3/n))) ∙e^(−nln(1+(1/n))) =lim_(n→∞) e^(n((3/n))) ∙e^(−n((1/n))) =e^(3−1) =e^2 , ∵ ln(1+x)∼_(x→0) x

Commented bymathdanisur last updated on 05/Aug/21

Ser, thank you

Commented bymindispower last updated on 06/Aug/21

(1+(a/n))^n →e^a

Commented byAr Brandon last updated on 06/Aug/21

Oh thanks !

Commented bymathdanisur last updated on 06/Aug/21

Thankyou Ser

Answered by ArielVyny last updated on 06/Aug/21

lim[((n(1+(3/n)))/(n(1+(1/n))))]^n =lim(((1+(3/n))^n )/((1+(1/n))^n ))=(e^(nln(1+(3/n))) /e^(nln(1+(1/n))) )=(e^3 /e^1 )=e^2

Commented bymathdanisur last updated on 06/Aug/21

Thankyou Ser

Answered by Olaf_Thorendsen last updated on 06/Aug/21

lim_(n→∞) (((n+3)/(n+1))) = ((lim_(n→∞) (1+(3/n))^n )/(lim_(n→∞) (1+(1/n))^n )) = (e^3 /e) = e^2

Commented bymathdanisur last updated on 06/Aug/21

Thankyou Ser