L'hopitals rule applied on probability density functions

Question

I'm going through a set of solutions for the following question

Let $Z_1,Z_2,\ldots$ be IID random variables with density $f$. Suppose that $P(Z_i>0)=1$ and that $\lambda = \mathrm{lim}_{x \rightarrow 0} f(x)>0$. Let $X_n = n \mathrm{min}\{Z_1,\ldots,Z_n\}$. Show that $X_n$ converges in distribution to $Z$ where $Z$ has an exponential distribution with mean $\frac{1}{\lambda}$.

I understand the solution up until it claims using L'Hopital's rule that $1-e^{\mathrm{log}(1-F(x/n))/\frac{1}{n}} \rightarrow 1-e^{-x\lambda}$. How can i prove this intermediate step with L'Hopitals rule?

Answer 1

It's easy to know that L'Hospital's rule can be applied. And then \begin{align*} \lim_{n\to\infty}\frac{\log(1-F(x/n))}{1/n}=\lim_{n\to\infty} \frac{f(x/n)\frac x{n^2}}{-\frac1{n^2}(1-F(x/n))}=-x\lim_{n\to\infty}\frac{f(x/n)}{1-F(x/n)}=-x\lambda. \end{align*}