Proving the limit of a given cubic function from the epsilon-delta definition of a limit

Question

Would anyone know how to prove the limit of this cubic equation using the epsilon delta definition?

$\lim_{x \rightarrow 2} x^3 +2x^2 -x -1 = 13 $ I really don't know where to start other than inputting the values of $a$, $L$ and $f(x)$ for this example into the definition of a limit:

$0 < |x - 2| < d $ implies $|(x^3 +2x^2 -x -1) - 13| < \epsilon$

Answer 1

Let $f(x)=x^3+2x^2-x+1. $ Note $|f(x)-13|=|x-2||x^2+4x+7|$

For $0<|x-2|<1$, we have $1<x<3\Rightarrow12<x^2+4x+7<28$,

Let $\epsilon>0$, then take $\delta(\epsilon):=\min\{\epsilon/28,1\}$. For $0<|x-2|<\delta(\epsilon), $ we have

$$|f(x)-13|=|x-2||x^2+4x+7|<\frac{\epsilon}{28}28=\epsilon$$