After doing a lot of homework, I've realized that I don't really get how to use the $\left(1+\frac{1}{n}\right)^n\to e$ method, which shows in how I can't solve indeterminate limits of the type $1^\infty$. For example:
$\lim_{n\to\infty}{\left(\frac{1}{2}+\frac{2}{n}\right)^n}$
Then I would do as follows:
$\lim_{n\to\infty}{\left(\frac{1}{2}+\frac{1}{2}+\frac{2}{n}-\frac{1}{2}\right)^n}$
$\lim_{n\to\infty}{\left(1+\frac{4-n}{2n}\right)^n}$
$\lim_{n\to\infty}{\left(1+\frac{1}{\frac{2n}{4-n}}\right)^n}$
And, from here on, I would not know what to do. Thus, the question is:
How am I supposed to work with the exponents?