Solve Epsilon & Delta for Homework Statement

[goodz]

Homework Statement

For the limit below, find values of δ that correspond to the ε values.

Homework Equations

epsilon = .5
and
epsilon = .05

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

 

[Hurkyl]

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.

 

[D H]

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

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.

i get |x-1|<0.2619

but its incorrect

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

 

[vela]

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.