Question

Find the set of δ values that satisfy the formal definition of lim(11/g(x))=2.2 as x approaches 4 when given the value ε = 0.5 and g(x) = x2 – x – 7, showing all work.

Answer 1

Do you know the formal delta-epsilon definition of a limit?

Answer 2

Yes. I have gotten to the following:

Let g(x)=x^2-x-7, L=2.2, a=4, and ε=0.5
Then, we need to prove that there exists a δ such that |11/(x^2-x-7)-2.2|<0.5 for all 0<|x-4|<δ.

-0.5<11/(x^2-x-7)-2.2<0.5
-0.5<11/(x^2-x-7)-2.2((x^2-x-7)/(x^2-x-7))<0.5
-0.5<11/(x^2-x-7)-(2.2x^2-2.2x-15.4)/(x^2-x-7)<0.5

Answer 3

I'm not sure where to go from here

Answer 4

-0.5<11/(x^2-x-7)-2.2<0.5
1.7 < 11/p(x) < 2.7
p(x)<11/1.7 and p(x)>11/2.7.

Answer 5

Where did the p(x) come in?

Answer 6

look at the the intersection of both sets of possble x's. Then look at the biggest delta such that x-detla, x+delta is in this intersection.

p(x) = x^2-x-7

Answer 7

I am still confused as to how you got each step.

Answer 8

first step: adding 2.2 to each term.
second step: a < b < c is actually a shortcut for a < b and b < c.

1.7 < 11/p(x) is equivalent to p(x) < 11/1.7 (it's ok because p(x) can't be allowed to be negative since (11/'negative number') < 0)

also, 11/p(x) < 2.7 is eqvlt to p(x) > 11/2.7
(here it's false.. it's actually p(x) > 11/2.7 OR p(x) < 0, but we know that we can't allow the negative values for p(x)..)

Answer 9

first inequality fives as solution set S1, second inequality gives S2. You take S = S1 intersection with S2. And find the biggest delta such that ]4-delta,4+delta[ is in S. (i didn't do it, but i tihnk it will be like that)

Answer 10

gives*

Answer 11

So after getting p(x)<11/1.7 and p(x)>11/2.7, i choose which one has the larger value for delta? That is, 11/1.7=6.470588235 and 11/2.7=4.074074074.... which tells that delta is 11/1.7?
I guess that is where I am confused now

Answer 12

nope. solve p(x) < 11/1.7 (doesn't look great)
=> solutions form a set S1.
solve p(x) > 11/2.7
=> solutions form a set S2.

Answer 13

Sorry.... I am confused on what you mean by solutions from a set S1 and S2...

Answer 14

example: the solutions of x^2-1 < 3 are the solutions of x^2 < 4. solution set: ]-2,2[. this is S1.

Answer 15

Ok... so now i need to take p(x)=x^2-x-7 < 11/1.7 and solve :)

Answer 16

and do the same for p(x)>11/2.7

Answer 17

yes. then \(S=S_1 \cap S_2\). and look for that \(\delta\).

Answer 18

Ok... would you mind if I work this out then send you a private message to see if I did it correctly?

Answer 19

quickly then ;)