Question
Given $f : \mathbb{R}$ $\rightarrow$ $\mathbb{R}$ a continuous function satisfying the functional equation : $$ f(x+y) - 3^y f(x) = 3^x f(y)$$ for all $x,y$ $\in \mathbb{R}$.
Let $$L = \lim_{x\to \infty} \left(\frac{f(x)}{f'(x)} + \frac{f'(x)}{f''(x)}+\cdots+ \frac{f^{n-1}(x)}{f^n(x)}\right)$$
Find L
My approach:
I could not find $f(x)$ itself. Usually, my approach is partially differentiating and setting one variable to zero, but it does not work in this case.
$$f'(x+y) - 3^y f(x) =(\ln3) 3^x f(y) $$
Setting $y=0$ simply gives $0=0$.
Any hints would help a lot. Thank you