For the limit

Question

lim
x → 2
(x3 − 5x + 3) = 1
illustrate the definition by finding the largest possible values of δ that correspond to ε = 0.2 and ε = 0.1. (Round your answers to four decimal places.)

i set it up like this

1-0.2</x^3-5x+3/<0.2+1
0.8</x^3-5x+3/<1.2

then I replaced 0.8 and 1.2 in tho the function and f(0.8)= -0.488
f(1.2)=-1.272

do i add and subtract these from 2?

Answer 1

For ε=0.1, you want to find δ such that

|f(2+δ)-f(2)| < 0.1
|((2+δ)^3-5(2+δ)+3)-(2^3-5*2+3)| < 0.1
|δ^3+6δ^2+7δ| < 0.1
δ < 0.014