Question

Having trouble with an epsilon-delta problem. Please help!

Answer 1

Here is the problem. I tried isolating x to start, the problem was getting more complicated than I thought it should be. Any ideas?

Answer 2

\(|p(x) - p(1.9) |=|(1-1)+2(x-1.9) + 3(x-1.9)(x+1.9)| = |x-1.9|\; |2 + 3(x+1.9)|.\)

Answer 3

I think you can solve the problem with this hint.

Answer 4

If you didn't solve yet, see this hint. Since \( |2 + 3(x+1.9)| = |3(x-1.9) + 13.4| \ge 3|x-1.9|+13.4\) and the equality holds if \(x \ge 1.9\). Supposing that \(x \ge 1.9\), \[|p(x) - p(1.9)| = |x-1.9| (3|x-1.9| + 13.4) < \delta (3\delta + 13.4). \] So if delta has its maximum, we have \(\delta(3\delta + 13.4) = 0.00025,\) that is \(\delta = 0.0000186566.\) (Wolfram Alpha.)