Evaluate the limit (if it exists) $\{(\frac{n^3+4n-5}{n^6+2n^2-3},n-7[\frac{n}{7}])\}_{n=1}^{\infty}$ 
$$\left\{\frac{n^3+4n-5}{n^6+2n^2-3},n-7\left[\frac{n}{7}\right]\right\}_{n=1}^{\infty}$$
where $\left[\frac{n}{7}\right]$ is the floor and ceiling function

The sequence converges if each element convarges so:
$$\lim_{n\to \infty} \frac{n^3+4n-5}{n^6+2n^2-3}=\lim_{n\to \infty} \frac{\frac{n^3}{n^6}+\frac{4n}{n^6}+\frac{-5}{n^6}}{\frac{n^6}{n^6}+\frac{2n^2}{n^6}+\frac{-3}{n^6}}=0$$
$$\lim_{n\to \infty}n-7\left[\frac{n}{7}\right]=\infty\cdot \infty=\infty$$
So $$\left\{\left(\frac{n^3+4n-5}{n^6+2n^2-3},n-7\left[\frac{n}{7}\right]\right)\right\}_{n=1}^{\infty}=\{0,\infty\}$$
And therefore there is not limit?

The remainder-like function
$$
n-7\left[\frac n7\right]
$$
cycles among the values $\{0,1,2,3,-3,-2,-1\}$. Since one of the components does not converge, the sequence of pairs does not converge.
