$\lim\limits_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n(1-\cos(1/n^2))}$ I want to solve this limit:
$$\lim_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n(1-\cos(1/n^2))}$$
I have proved that $\lim\limits_{n \rightarrow +\infty} \frac{\ln(1+n+n^3)-3\ln(n)}{n} = 0$ and $\lim\limits_{n \rightarrow +\infty} \frac{1}{(1-\cos(1/n^2))}= \infty$ but I have indeterminate form. How can I solve that?

Let's write
$$\frac{\ln(1+n+n^3) - 3 \ln(n)}{n(1- \cos(1/n^2))} = \frac{\ln(1 + \frac{1}{n^2} + \frac{1}{n^3})}{n(1- \cos(1/n^2))} = \frac{\frac{1}{n^2} + \frac{1}{n^3} +o(\frac{1}{n^3})}{n (\frac{1}{2n^4} + o(\frac{1}{n^5}))} = \frac{n +1 + o(1)}{\frac{1}{2} + o(\frac{1}{n})} \sim 2n$$
so the limit is $+\infty$.
