$\lim\dfrac{n^3-2}{n+1}$
$=\lim\dfrac{n^3\Big(1-\dfrac{2}{n^3}\Big)}{n\Big(1+\dfrac{1}{n}\Big)}$
$=\lim n^2.\dfrac{1-\dfrac{2}{n^3}}{1+\dfrac{1}{n}}$
$=+\infty$
$\lim\dfrac{1+3^n}{4+3^n}$
$=\lim\dfrac{\dfrac{1}{3^n}+1}{\dfrac{4}{3^n}+1}$
$=\dfrac{0+1}{0+1}$
$=1$
