Question: I would like to prove that $$\lim\_{n\to\infty} \sqrt[3]{(n+1)(n+2)(n+3)}-n = 2,$$
but have no ideas on how to proceed with this limit. Any hint? Thanks in advance

Answer: 
Here is a hint.
$$a-b=\frac{a^3-b^3}{a^2+ab+b^2}$$
Hint : factorize by $n$ and use a Taylor expansion of $f(t)=(1+t)^\frac{1}{3}$ when $t$ is near zero.