How do I solve this problem

Question

In formally proving that , let  be arbitrary. Choose min. Determine the least possible whole number  such that delta is of the indicated form.

Answer 1

Let ε > 0 arbitrary number choose ð = min(ε/m, 1)

|x- 4| < ð |x³ - 4x² + x - 4| = |x - 4|(x² + 1) <ð(x² + 1) = ε; ð = ε/(x² + 1);

Let ð = 1; |x - 4| < 1; - 1 < x - 4 < 1; 3 < x < 5; 10 < x² + 1 < 26

ð = min(ε/26, 1), o m = 26

Answer 2

Let f(x)=x³-4x²+x

f(4+h) = 4 + 17h - 8h² + h³

epsilon<h(17-8h+h²)

For h near 0 the quadratic factor near is less than 18 but greater than 14.

Therefore if h<epsilon/18, 17h-8h²+h³ will be less than epsilon, i.e. delta=epsilon/18