I was doing an exercise about limits of sequences and arrived at the following limit: $$\lim_n \left(1+\frac{1}{-n} \right)^{-n}\ \ \ \ (1)$$
We are supposed to solve the limit without using L'hopital's rule. The only limit that's similar to this one that I know is:
$$\lim_n \left(1+\frac{1}{n} \right)^n = e\ \ \ \ (2)$$
But I have no clue how to get to something similar to $(2)$ starting from (1). How can this be done?
I assume you're taking $n\to\infty$ in these limits. We have $$\lim_{n\to\infty}\left(1-\frac1n\right)^{-n}=\lim_{n\to\infty}\left(\frac n{n-1}\right)^n=\lim_{n\to\infty}\left(1+\frac1{n-1}\right)^n$$ by algebra. Clearly this equals $$\lim_{n\to\infty}\left(1+\frac1n\right)^{n+1}$$ after replacing $n$ by $n+1$. I leave it to you to proceed from here by splitting this into two pieces.