# Epsilon and delta

1. Jan 31, 2010

### goodz

1. The problem statement, all variables and given/known data
For the limit below, find values of δ that correspond to the ε values.

2. Relevant equations
epsilon = .5
and
epsilon = .05

3. The attempt at a solution
These kinds of problems do you have to use a graphing calculator to figure it out?
for epsilon = .05
|(9x + x - 3x^3)-7|<.05
6.95<(9+x-3x^3)<7.05

and I graph it, i get x = about -.75548, y=6.95
x = -.73803, y=7.05

i get |x-1|<0.2619

but its incorrect

2. Jan 31, 2010

### Hurkyl

Staff Emeritus
Algebra could be used, couldn't it? Isn't solving quadratic inequalities one of the things they teach in pre-calculus?

(I haven't checked your arithmetic, I'm assuming it's right)

Anyways, you are misunderstand something. You did discover* that the interval** (-7.55, -7.39) does have the property that, for every x in it, f(x) lies within the interval (6.95, 7.95).

But that interval is not described by the inequality |x-1|<0.2619....

*: Well, more precisely, you have some evidence to suggest it. To really be confident in it, you have to find some algebraic proof, or a deeper understanding of approximations and conic functions.
**: Of course, you found a slightly larger interval, but the difference isn't really relevant.

Last edited: Jan 31, 2010
3. Jan 31, 2010

### D H

Staff Emeritus
Try again! First off, those values are nowhere near 1. That should have been a first hint. Secondly, those values do not yield anything close to 7. The values for 9+x-3x3 for x=-0.75548 and x=-0.73803 are 2.538 and 2.438.

You made another mistake here. -0.75-1=-1.75, not 0.25.

4. Jan 31, 2010

### vela

Staff Emeritus
Try considering three separate limits and make use of the triangle inequality to find a delta that'll work for a given epsilon. That way you won't have a hard cubic to solve.