finding the limit as n goes to infinity

Question

I'm just trying to find the result of $$\lim_{n \to \infty} \frac{f(n)}{g(n)}$$ to see if $f(n)$ is $O(g(n)), \sigma(g(n)), \Theta(g(n))$,

where $f(n) = log n + \log n$

and $g(n) = log \log n$

Can someone explain that to me or at least find out the result of

$$ \lim_{n \to \infty} \frac{ \log n}{10 n + \log \log n}? $$

thanks

Answer 1

Use equivalents: $$\left.\begin{array}{l}f(n)\sim_\infty50 n\log n\\ g(n)\sim_\infty 10 n,\end{array}\right\}\enspace\text{hence}\enspace\frac{f(n)}{g(n)}\sim_\infty \frac{50n\log n}{10 n}=5\log n\xrightarrow[n\to\infty]{}+\infty.$$