How to solve this limit? (Without L'Hôpital's rule)

Question

I only know that it should be something with $\,e\,$ but I don’t have and idea how to solve it. $$\lim_{n\to\infty}\frac{(n+1)^n-(n+2)^n}{(n+2)^n-(n+3)^n}$$

Answer 1

Write as $\;\lim\limits_{n\to\infty }\dfrac{\big[(n+1)/(n+2)\big]^n-1}{1-\big[(n+3)/(n+2)\big]^n}\;$.

Answer 2

All you need is $\;\lim_\limits{n\to\infty}\left(1+\dfrac1n\right)^n=e\,$.

From this you can get $\;\lim_\limits{n\to\infty}\left(1+\dfrac{2}{n}\right)^n\,$ and $\;\lim_\limits{n\to\infty}\left(1+\dfrac{3}{n}\right)^n\;$ which will get you the result.