<s>[INST] Solving limit for $\lim_{n \to \infty} \frac{3n^3 + 5n^2 + (-1)^n \cdot n}{5n^2 - 13}$ Trying to solve:
$$\lim_{n \to \infty} \frac{3n^3 + 5n^2 + (-1)^n \cdot n}{5n^2 - 13}$$ 
I tried to divide with $n^3$ and $n^2$, but the limit of $(-1)^{n}$ is not existed, so i am wondering if the limit of $(-1)^{n}\cdot n$ does exist ? 
 [/INST] We have indeed that $\lim_{n \to \infty} (-1)^{n}\cdot n$ doesn't exist but we have that
$$ \frac{3n^3 + 5n^2 + (-1)^n \cdot n}{5n^2 - 13} =  \frac{n^3}{n^2}\cdot\frac{3 + \frac 5 n +\frac{(-1)^n}{n^2} }{5- \frac{13}{n^2}}$$
and
$$\left|\frac{(-1)^n}{n^2}\right|= \frac{1}{n^2}\to 0 \implies  \frac{(-1)^n}{n^2}\to 0$$
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