Finding the limit of an integral

Question

Evaluate $$\displaystyle\lim_{j\rightarrow \infty} \displaystyle\int_{0}^{a} \frac{1}{j!} \left(\ln \left(\frac{A}{x}\right)\right)^{j}dx$$

Answer 1

I'll take $j=n$, as an integer.

Using IBP, we may show that the integral would be \begin{align*} \int_{0}^{a} \frac{1}{n!} \left(\ln \left(\frac{A}{x}\right)\right)^{n}dx &= a\, \sum_{k=0}^{n} \frac{\left(\ln{\frac{A}{a}}\right)^k}{k!} \\ \implies \lim_{n\to\infty} \int_{0}^{a} \frac{1}{n!} \left(\ln \left(\frac{A}{x}\right)\right)^{n}dx &= a\, e^{\ln{(A/a)}} \\ &= a\cdot \frac{A}{a} \\ &= A \end{align*}