<s>[INST] Prove a limit for $g(x)$ with the definition of limit only We have the function $g(x) = x^3+1$
Prove, with the definition of limit only, that the limit $L=9$ is indeed the limit of the function when  $x=2$.
I started with: $\lvert g(x) - L \rvert$ : $\lvert x^3+1-9 \rvert = \lvert x^3-8 \rvert = \lvert (x-2)(x^2+2x+4) \rvert < \epsilon $
From the definition of the limit we know: $ \rvert x-2 \lvert < \delta$,  so therefore: $\lvert (x-2)(x^2+2x+4) \rvert < \delta (x^2+2x+4) $
How do I proceed from here? do I need to find a condition for $ x^2+2x+4 $ ?
Thanks guys 
 [/INST] $$|x^2+2x+4|=|x^2-4x+4+6x-12+12|=|(x-2)^2+6(x-2)+12| \leq |\delta^2+6\delta+12|$$
So for small $\delta$ you have (for example) $|x^2+2x+4|<13$
</s>