Question

For the limit
lim (x^3 − 2x + 8)=12
x → 2
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.)

ε = 0.2 δ =?
ε = 0.1 δ =?

Answer 1

So we need to find \( \delta \) in \(|x-2|< \delta \) so that \(|f(x)-12|<0.2\)
I thought we could simplify it like this |x3-2x-4|<0.2
Then from the condition on \( \delta \) get this inequalty \(|2(\delta-2)+(\delta+2)^{2}-4|<0.2 \) and from solving it get the largest possible values of δ.

Answer 2

I just completed the problem here was my process:
\[-0.2<(x ^{3}-2x+8)-12<0.2\]
add 12 compound inequality.
\[11.8<x ^{3}-2x+8<12.1 \]
then i determined the values of x for which the curve \[y=x ^{3}-2x+8\] lies between the horizontal lines y=11.9 and y=12.1.
I then calculated the intersections which were @11.8 x=1.9798 @12.2 x=2.197
Then subtracted 2 from both and grabbed the smallest value to reach my delta.

Answer 3

so my answers were:
\[\epsilon=0.2 \delta=.0197\]
\[\epsilon= 0.1 \delta=.0099\]