<s>[INST] How to compute this gross limit. 
How do I compute this limit?
  $$
  \lim_{n \to \infty} 
  \frac{\left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - 
        \left(1 + \frac{1}{n} - \frac{1}{n^2}\right)^n
  }{
       2 \left(1 + \frac{1}{n} + \frac{1}{n^2}\right)^n - 
        \left(1 + \frac{1}{n} - \frac{1}{n^2 + 1}\right)^n - 
        \left(1 + \frac{1}{n} - \frac{1}{n^2 (n^2 +1)}\right)^n
  }
$$

I think I got the correct limit by using fast converging limits to $e$.
In particular I used truncated Taylor series for the sqrt and 4th root.
Or squares and bisquares.
Example
$(1+1/2n)^{2n}$
 Becomes 
$(1 + 1/n + 1/4n^2)^n.$
In combination with l'hopital it gives me the answer.
But I guess that is not a very good (fast) method.
 [/INST] Hint: (for the first version, before multiplying by 2)
$$
\left(1+\frac1n\right)^n\leq\left(1+\frac1n+\frac1{n^2}\right)^n\leq\left(1+\frac{1+\epsilon}n\right)^n
$$
for large $n$. Hence all partial limits are $e$ and the final one — 0.
</s>