Question

Given the lim(3x-7)=2 as x-->3. Using the epsilon-delta definition for the given epsilon=.01 find the corresponding delta > 0 such that |(3x-7) - 2| < .01 whenever |x-3| < delta

Answer 1

So when \(\large\rm \epsilon=0.01\), there exists a \(\large\rm \delta\gt0\) such that\[\large\rm 0\lt|x-3|\lt\delta\qquad\implies\qquad |(3x-7)-2|\lt0.01\]

So start with your epsilon equation, trying to find an x-3 in there.

Answer 2

\[\large\rm |3x-9|\lt0.01\]Do you see it? :) It's so close!!

Answer 3

X < 3.003?

Answer 4

sorry I suck at math lol

Answer 5

Well you're not trying to solve for x in this inequality! :O careful!\[\large\rm \color{orangered}{|x-3|}\lt\delta\]You're trying to make this orange part show up in your epsilon equation.

Answer 6

So if I factor out a 3,\[\large\rm |3(\color{orangered}{x-3})|\lt0.01\]

Answer 7

And divide the 3 over to the other side,\[\large\rm \color{orangered}{|x-3|}\lt\frac{1}{300}\]

Answer 8

If you compare these two inequalities:\[\large\rm \color{orangered}{|x-3|}\lt\delta\qquad\qquad\qquad \color{orangered}{|x-3|}\lt\frac{1}{300}\]It looks like we've found our delta, do you see it? :o

Answer 9

I rewrote 0.01 as 1/100 and then divided by 3, or multiplied by 1/3,
just in case that middle step was confusing.

Answer 10

cee peeeeeee +_+ where you at broski?
brain esplode?

Answer 11

thanks so much, i understand it a little better now

Answer 12

ya these are super weird :U
calc gets a lot more fun once you get past this initial stuff

Answer 13

i hope so lol

Answer 14

can you help me with one more?

Answer 15

sure!

Answer 16

thanks!given the epsilon-delta definition prove that lim(3x-5)=7 as x --> 4

Answer 17

so would it become |x-4| < \[\epsilon \div 3\]

Answer 18

so delta would be epsilon/3

Answer 19

Mmm ya that sounds right! :)
If they say "prove" then maybe we need to write it up formally, with some words and stuff...

Answer 20

okay thanks!

Answer 21

\[\large\rm \lim_{x\to4}3x-5=7\]
Let \(\large\rm \epsilon\gt0\).
There exists a \(\large\rm \delta\gt0\) such that
\[\large\rm \text{if}\quad 0\lt|x-4|\lt\delta\qquad\text{then}\qquad |f(x)-7|\lt\epsilon\]

\[\large\rm |f(x)-7|\lt\epsilon \quad\text{if and only if}\quad |3x-5-7|\lt\epsilon\]\[\large\rm \text{if and only if}\quad|3x-12|\lt\epsilon\]\[\large\rm \text{if and only if}\quad|x-4|\lt\frac{\epsilon}{3}\]

Choose \(\large\rm \delta=\dfrac{\epsilon}{3}\).
Thus, \(\large\rm 0\lt|x-4|\lt\dfrac{\epsilon}{3}\) and it follows that \(\large\rm |f(x)-7|\lt\epsilon\).

Maybe something like that? :p
Bah I can't remember how to write these properly.

Answer 22

But ya you've got the right idea with finding the corresponding delta value :D

Answer 23

The first type of problem is a little easier to deal with, you're just looking for a specific number.

In the second problem you're generalizing for every epsilon, so you gotta be careful :O

Answer 24

Nothing to add, you've done a fine job @zepdrix :)

Answer 25

What do you think cee pee? :o
Too much with the words?
It's just each step following the previous step.

Relating delta to the epsilon/3 is really the important part though, which it seems like you've figured out.