Consider the sequence $c_m=\frac{m-4}{2m+1}$. Does this sequence converge? If so, to what value?

Evaluate the limit of the sequence $c_n={\tan }^{-1}\left(\frac{12n}{12n+6}\right)$ as $n\to \infty$.

Evaluate the limit of the sequence $\left\{b_n\right\}$ defined by $b_n=\sqrt{n^2+4n}-n$ as $n\to \infty$, or state if the sequence diverges.

Consider the sequence given by $c_n=\frac{(2n^2+5n-3)\sin n}{7n^3-n}$. Using the Squeeze Theorem, determine $\underset{n\to \infty }{\lim }{c}_n$.

Find the limit of the sequence $b_n=\frac{5n^4+2}{3n^4-7}$ as $n\to \infty$.