<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] $y = x+2$:
$$L = \lim _{x\to \infty} (\sqrt[3]{(x+1)(x+2)(x+3)}-x) = 2+ \lim _{y\to \infty} (\sqrt[3]{y^3-y}-y) = 2+ \lim _{y\to \infty} \frac{\sqrt[3]{1-\frac{1}{y^2}}-1}{\frac{1}{y}}$$
$y = \frac{1}{h}$:
$$L = 2+ \lim _{h\to 0^+}\frac{\sqrt[3]{1-h^2}-1}{h} \stackrel{L'Hopital}{=}2+ \lim _{h\to 0^+}\frac{\frac{-2h}{\sqrt[3]{(1-h^2)^2}}}{1} = 2+0 = 2$$
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