Proving a cubic limit with the epsilon/delta definition of a limit

Question

So at first I was asked "if $\lim \limits_{x \to a} f(x) = L$ then prove $\lim \limits_{x \to a} f(x)^2 = L^2$". In order to do this I simply just proved the product rule of limits. The second part of the question gets more complicated as it asks "if $\lim \limits_{x \to a} f(x) = L$ then prove that $\lim \limits_{x \to a} f(x)^3 = L^3$. You must use the identity $a^3−b^3=(a−b)(a^2+ab+b^2)$ in your proof". I don't understand where to go with this question. I started off by making $|f(x)^3-L^3|$ into $|(f(x)-L)(f(x)^2+f(x)L+L^2)|$ but I have not a single clue what to do with this. If anyone might have an idea of what to do, then helping would be appreciated very much. Thank you.

Answer 1

Let $0 <\epsilon <1$. Then $$|(f(x)-L)(f(x)^{2}+f(x)L+L^{2})|$$ $$ \leq |(f(x)-L)| ((|L|+\epsilon)^{2}+(|L|+\epsilon)L+L^{2}$$ $$<|(f(x)-L)| ((L+1)^{2}+L(|L|+1)+L^{2})$$ $$ <\epsilon$$ if $\delta$ is so small that $|f(x)-L)| <\epsilon /M$ for $|x-a| <\delta$ where $M=(|L|+1)^{2}+|L|(L+a)+L^{2})$.

Answer 2

Hint: Use the identity $$a^2+ab+b^2=(a-b)^2+3ab.$$