How to prove this limit using definition? How can we prove that
$$\lim_{x\rightarrow 3} \left ( x^{3} - 3x + 2 \right ) = 20$$
using the definition with $\epsilon$ and $\delta$?

You have to show $$|x^{3}-3x + 2 -20| < \epsilon \qquad \text{whenever}\ \ \ \ |x-3| < \delta$$ $$\Longrightarrow |x^{3} -3x -18| = |(x-3)| \cdot  |x^{2}+3x+6| \qquad \text{whenever}\ \ \ \ |x-3| < \delta$$
Can you do it from here.
