Problem with estimating a sequence with intuition

When you have$$\lim_{n\to\infty}\frac{\ln(n)}{\ln(4n)},$$the denominator doesn't increase at a much faster rate than the numerator. As a matter of fact, since we have$$(\forall n\in\mathbb N):\ln(4n)=\ln(4)+\ln(n),$$they increase at the same rate. And now it is easy to see that the limit is indeed $1$.

Note that $\ln(4n)=\ln(4)+\ln n$ so that this function grows at the same rate as $\ln n$. In particular $$ \lim_{n\to \infty} \frac{\ln(n)}{\ln(4n)}= \lim_{n\to \infty} \frac{\ln(n)}{\ln(4)+\ln(n)}=\lim_{n\to \infty} \frac{1}{\frac{\ln(4)}{\ln (n)}+1}=1. $$ Also note that your intuition can be formalized. If $a_n\sim b_n$ and $c_n\sim d_n$ (where $\sim$ means that the ratio of the two sides goes to one), then $$ \lim_{n\to \infty}\frac{a_n}{c_n}=\lim_{n\to\infty}\frac{b_n}{d_n}. $$