I’m trying to understand the reasoning in the following step of a limit analysis:

$$lim_{n to infty} nleft(left(1- frac{1+c}{frac{n}{ln(n)}} right)^{frac{n}{ln (n)}}right)^{(n-1)ln n/n} = lim_{n to infty} ne^{-((1+c)ln(n))}$$

I understand the “inner” part; $lim_{n to infty} (1-frac{1+c}{frac{n}{ln (n)}})^{frac{n}{ln n}} = e^{-(1+c)}.$ And I sort of see that outer exponent $((n-1)ln n) /n = (ln n) – (ln n / n)$ and the second part goes to 0, but it’s not clear to me what rules actually justify “bringing the limit to the exponent”. What are the actual steps involved in deducing this limit?

More generally, these types of asymptotic analysis show up in comp sci all the time and I feel there is a bag of tricks that I am missing.