Find the limit of the sequence a_{n}=((2n+1)/(2n+4))^(n^2/(n+1))

Find the limit of the sequence a_{n}=((2n+1)/(2n+4))^(n^2/(n+1))
Thanks for your help

lim [(2n+1)/(2n+4)]^(n²/(n+1))
n->

this has a form of Euler's limit:

lim ( 1 +1/n)^n =e
n->∞

(2n+1)/(2n+4) =1 +1/y

(2n+1)y=(2n+4)y+2n+4
3y=-2n-4
y=(-2n-4)/3

(2n+1)/(2n+4) = 1 +1/[(-2n-4)/3]

If we multiply 3n²/(-2n-4)(n+1) by (-2n-4)/3
then we will get the original exponential that is n²/(n+1)

lim { [1 +1/[(-2n-4)/3]^[(-2n-4)/3] }^[3n²/(-2n-4)(n+1)]}
n->∞

lim e^[3n²/(-2n-4)(n+1)]
n->∞

lim e^[3n²/(-2n²-6n-4)]
n->∞

lim e^[3n²/-2n²[-1 +3/n +2/n²]]
n->∞

lim e^[-3/2(-1 +3/n +2/n²)]
n->∞

as lim n->∞ then

e^(-3/2(-1 +3/∞ +2/∞²) = e^(-3/2) or 1/e^(3/2)

the limit is 1/e^(3/2)