Given:
$$f\left(x\right) = a\ln{\left( bx \right)}$$
Then:
$$f'\left(x\right) = \frac{a}{x}$$ $$f''\left(x\right) = \frac{-a}{x^2}$$ $$f'''\left(x\right) = \frac{2a}{x^3}$$ $$f''''\left(x\right) = \frac{-6a}{x^4}$$
In general, I think this is:
$$f^n\left(x\right) = \frac{a\left(-1\right)^{n-1}\left(n-1\right)!}{x^n}$$
When $n\rightarrow\infty$, can this limit be expressed in a simple way? The limit of:
$$\lim_{n \to \infty} \frac{a\left(-1\right)^{n-1}\left(n-1\right)!}{x^n}$$