Limit I don't know how to start solving. This is the limit:
$$\lim_{n\to\infty}\frac{2^{n-1}-4^n}{3^n\sin n + 4^{n+2}}$$
I have a solution and the steps and I still haven't understood how it's done, here's the proposed solution:
$$\lim_{n\to\infty}\frac{2^{n-1}-4^n}{3^n\sin n + 4^{n+2}}=\lim_{n\to\infty}\frac{\frac12(\frac24)^n-1}{(\frac34)^n\sin n+16}=-\frac1{16}$$
WolframAlpha says this is correct but I haven't understood from where did all the fractions come from...

As often, the simplest is to use equivalents:
$2^{n-1}=_\infty o(4^n)$, hence $2^{n-1}- 4^n\sim_\infty- 4^n $. Similarly $\;3^n\sin n+4^{n+2}\sim_\infty4^{n+2}$, whence
$$\frac{2^{n-1}- 4^n}{3^n\sin n+4^{n+2}}\sim_\infty\frac{- 4^n}{4^{n+2}}=-\frac 1{4^2}.$$
