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**hmmmm** so you have factored out x-a so now you want to use delta to put a bound on that, you then know what x will vary between and so you will know the max value of |x^2+x+1|, does that help?

$\displaystyle \displaystyle\mid(f(x)-L)$

$\displaystyle \displaystyle\mid\=\mid(x^3-1)\mid\=\mid(x-1)\mid\mid(x^2+x+1)\mid$

We then use $\displaystyle \displaystyle\delta$ to bound $\displaystyle \displaystyle\mid(x-1)\mid$

so let $\displaystyle \displaystyle\delta=1\rightarrow\mid(x-1)\mid\le1$ so we know that x varies between -1 and 2

so $\displaystyle \displaystyle\(x^2+x+1)\le7$