Evaluating $\lim_{n\to\infty}\left(\frac{n+3}{n+1}\right)^{2n}$ I am having a hard time finding the next step on this.
Can someone tell me what is the next step and explain to me
in non-mathematical terms what I am supposed to do when I reach
this examples at this place ?
\begin{align*}
\lim_{n\to\infty}\left(\dfrac{n+3}{n+1}\right)^{2n}
&=\lim_{n\to\infty}\left(\dfrac{n+1+2}{n+1}\right)^{2n}\\
&=\lim_{n\to\infty}\left(1+\dfrac{2}{n+1}\right)^{2n}\\
&=\lim_{n\to\infty}\left(1+\dfrac{2}{n+1}\right)^{?}
\end{align*}

HINT:


*

*$e=(1+\frac1x)^x$

*$(1+\frac{2}{n+1})^{2n}=(1+\frac{2}{n+1})^{\frac{n+1}{2}\cdot4-2}=\frac{\left((1+\frac{2}{n+1})^{\frac{n+1}{2}}\right)^4}{(1+\frac{2}{n+1})^2}$
OK, apparently some more hints are required, so here is another one:
$$\lim\limits_{x\to\infty}\frac{f(x)^p}{g(x)^q}=\frac{\left(\lim\limits_{x\to\infty}f(x)\right)^p}{\left(\lim\limits_{x\to\infty}g(x)\right)^q}$$
