points$)$ Identify the specific indeterminate form, before discussing the convergence of the following sequences, and providing their limits$,$ if existent.

i$.,a_n$:$=(1+\frac{4}{n+12}{)}^n$

ii$.b_n$:$=\frac{\sqrt[\phantom{2}]{n}-3n}{\sqrt[3]{n}-1}$

iii. $c_n$:$=n(\sqrt[\phantom{2}]{9+\frac{2}{n}}-3)$

iv$.**d_n$:$=\frac{3n-3sin(n^3)}{n+1}$