<s>[INST] Evaluating $\lim _{x\to \infty} (\sqrt[3]{(x+1)(x+2)(x+3)}-x)$ 
Evaluate:  $\lim _{x\to \infty}( \sqrt[3]{(x+1)(x+2)(x+3)}-x)$

I set $y= x-3$ for simplification and then tried to solve $y\to \infty$. 
I tried to use the tayor expansion of the cubic function f(y) in the cuberoot but that didn't help. 
How do I approach this problem then? 
 [/INST] Using general Binomial expansion, $((x+1)(x+2)(x+3))^\frac{1}{3}-x = ((x+2)^3-(x+2))^\frac{1}{3}=(x+2)-\frac{1}{3}(x+2)^{-2}(x+2)-\frac{1}{9}(x+2)^{-5}(x+2)^3+..-x=2-P(x+2)$, 
where the power of the polynomial $P(x+2)$ is negative. Letting $x \to \infty$, we get the limit 2.
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