Limit of sequence using Euler's sequence

Question

$\lim_{n\to\infty }\left(1+\frac{1}{2n+3}\right)^n$

I know that this approaches $e^{1/2}$ however don't know how to prove this. Any hints are appreciated.

Answer 1

$$\lim_{n\to\infty}\left(1+\dfrac1{an+b}\right)^n=\left(\lim_{n\to\infty}\left(1+\dfrac1{an+b}\right)^{an+b}\right)^{\lim_{n\to\infty}\dfrac n{an+b}}$$

Answer 2

Hint: $\left(1+\frac{1}{2n+3}\right)^{2n+3}=\left(\left(1+\frac{1}{2n+3}\right)^{n}\right)^2\left(1+\frac{1}{2n+3}\right)^{3}$.

Answer 3

Simply put $$ 2n+3=m\iff n=\frac{m -3}{2} $$ Now we have $$ \lim_{n\to\infty}\left(1+\frac{1}{2n+3}\right)^n= \lim_{m\to\infty}\left(1+\frac{1}{m}\right)^{\frac{m -3}{2}} $$ and applying the usual rules of calculation of limits you get the sought for result.