Hello, I read the stickied delta epsilon guide at the top of the page and found it very helpful for finding an epsilon for a given delta. The guide represents delta = epsilon/M. If M has an x in it, we need to set a bound on delta in order to get an M that is a number without any x's in it.

My problem is, for example, proving that the limit of x^3 as x approaches 3 is 27. Here is what I have so far:

$\displaystyle |x-3| < \delta$

$\displaystyle |x^3 - 27| < \epsilon$

$\displaystyle (|x - 3|)(|x^2 + 3x + 9|) < \epsilon$

$\displaystyle (|x -3|) < \epsilon/(|x^2 + 3x + 9|)$

In the stickied article, it says that I need to set a delta (i.e, delta = 1) and then add things to my first equation so that $\displaystyle |x - 3|$ becomes $\displaystyle |x^2 + 3x + 9|$. If i do this, my M still ends up having an x in it because the only way to turn |x -3| into a quadratic equation is to multiply by x. I can solve delta epsilon proofs, but the cubic powers are throwing me off here. Any suggestions?