How to find the limit of
$\lim_{n\rightarrow \infty}n^{2}((1+\frac{p}{n})^{q}-(1+\frac{q}{n})^{p}), (p,q,n \in N)$
I found that it equals to $\lim_{n\rightarrow\infty}(\frac{p^{q}*n^{max(p,q)}}{n^{q-2}}-\frac{q^{p}*n^{max(p,q)}}{n^{p-2}})$
but this look's like wrong statement, because I couldn't find anything simple from it.
What kind of method would work for this one?
Note that: $$\left(1+\frac{p}{n}\right)^q = 1 + \binom{q}{1}\frac{p}{n} + \binom{q}{2}\frac{p^2}{n^2} + O\left(\frac{1}{n^3}\right)$$ $$\left(1+\frac{q}{n}\right)^p = 1 + \binom{p}{1}\frac{q}{n} + \binom{p}{2}\frac{q^2}{n^2} + O\left(\frac{1}{n^3}\right)$$
Subtracting and multiplying by $n^2$: $$a_n = \left[\binom{q}{2}p^2-\binom{p}{2}q^2\right] + O\left(\frac{1}{n}\right)$$ where $a_n$ is the term in the limit. Taking $n\rightarrow\infty$, $O(n^{-1})$ terms become zero, so the result is the terms in the square brackets.