I think this is a limit proof question, but I wasn't sure. Any help?
Use a graph to find the largest number $\delta > 0$ such that if $|x-1|<\delta$, then $|x^3-5x+6-2| < \varepsilon$ when $\varepsilon=0.2$
Here's the graph of $y = x^3-5x+6$ as well as $y=2$ and $y=2\pm\varepsilon$. Can you read off the answer?
If you find the first graph difficult to read, here's $y = |x^3-5x+6-2|$ together with $y=\varepsilon$.