From the first sight it may seem that this limit is equal to 1

Why? Because $\frac{n^2+n}{n^2+2n+2}$ has limit 1. Hence, $\lim 1^n = 1$. What is obviously wrong.

The second idea that came to my mind is that we can see that for any $n$ $\frac{n^2+n}{n^2+2n+2}$ is less than 1. But it does not mean that it's limit is equal to 0. In fact, I created a graph in Desmos and it shows that it's limit is approximately 0.368.

Probably it has something to do with $e$ but I couldn't find the connection.