What are the mechanics of finding delta in a limit problem?

  • Thread starter Saladsamurai
  • Start date
In summary, to find an open interval on which the inequality |f(x)-L|<\epsilon holds, we simply solve the inequality and then choose a delta such that |x-x_0|<\delta. In this case, we can see that the open interval is (3.99, 4.01) and we can choose a delta of 0.01 to ensure that the inequality is satisfied. By combining the given information and using algebra, we can see that delta = |x-x_0|<0.01.
  • #1
Saladsamurai
3,020
7
Okay Then! :smile: I am going to start with a simple problem here:

Given some function, a limit L, an xo, and some [itex]\epsilon[/itex]:

a) Find an open interval on which the inequality [itex]|f(x)-L|<\epsilon[/itex] holds. Then b) give a value for [itex]\delta>0[/itex] such that
for all x satisfying 0 < |x - x0| < [itex]\delta\Rightarrow |f(x)-L|<\epsilon[/itex].

f(x)=x+1
L = 5
xo=4
[itex]\epsilon[/itex]=0.01

a) To find an interval on which [itex]|f(x)-L|<\epsilon[/itex] holds, I simply solve the inequality:

[tex]|f(x)-L|<\epsilon[/tex]

[tex]-\epsilon<f(x)-L<\epsilon[/tex]

[tex]-\epsilon<x+1-5<\epsilon[/tex]

[tex]-\epsilon<x-4<\epsilon[/tex]

[tex]4-\epsilon<x<4+\epsilon[/tex]

[tex]3.99<x<4.01[/tex]

So there is my open interval, (3.99, 4.01), on which [itex]|f(x)-L|<\epsilon[/itex] holds.

Now I know that for part (b), delta must be 0.01.

But what how do we actually find [itex]\delta[/itex]? What are the mechanics of finding it.

For part (a) I solved an inequality; what did I do for part (b) to find delta?

Sorry if this is a little vague, I am not sure exactly how to word my question.
 
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  • #2
A better question to ask is what is delta? Delta is how sufficiently close x must be to x_0 to guarantee that |f(x) -L| < epsilon will be satisfied. Thus, we want [itex]|x-x_0| < \delta[/itex] or [itex]x_0 - \delta < x < x_0 + \delta.[/itex] In this case, [itex]x_0 = 4[/itex] so we want a delta such that [itex]4 -\delta < x < 4 + \delta[/itex].

Now compare this last set of inequalities to what you concluded in part a). How do we choose delta to ensure that |f(x) - L| is indeed less than epsilon?
 
  • #3
snipez90 said:
A better question to ask is what is delta? Delta is how sufficiently close x must be to x_0 to guarantee that |f(x) -L| < epsilon will be satisfied. Thus, we want [itex]|x-x_0| < \delta[/itex] or [itex]x_0 - \delta < x < x_0 + \delta.[/itex] In this case, [itex]x_0 = 4[/itex] so we want a delta such that [itex]4 -\delta < x < 4 + \delta[/itex].

Now compare this last set of inequalities to what you concluded in part a). How do we choose delta to ensure that |f(x) - L| is indeed less than epsilon?

So since we know that we need [itex]|x-x_0| < \delta[/itex] (1), we also know xo=4 (2), and we have an inequality that says 3.99 < x < 4.01 (3), we can simply 'combine' (1), (2), and (3) to yield ...err something. I need a moment to think about it. But, I think I see it now.
 
  • #4
I am still a little lost here :redface: sorry. What is the next step?

I know that 3.99<x<4.01. I also know that |x-xo|<[itex]\delta[/itex]. And xo=4.

How do I combine the 3 into something meaningful to find [itex]\delta[/itex] ?

Thanks
 
  • #5
Nevermind. I was forgetting to subtract 4 from ALL sides of the inequality.

Using the above we have:

3.99-4 < x-4 < 4.01-4

-0.01<x-4< 0.01
or
|x-4|<0.01 = delta
 

1. What are Epsilons and Deltas?

Epsilons and Deltas are types of brainwashed individuals in Aldous Huxley's novel "Brave New World". Epsilons are the lowest caste of society, while Deltas are slightly higher but still considered inferior to Alphas and Betas.

2. How are Epsilons and Deltas created?

Epsilons and Deltas are created through a process of hypnopaedia, or sleep-teaching. They are conditioned from birth to fulfill specific roles in society and have no individuality or free will.

3. What is the purpose of Epsilons and Deltas in society?

The purpose of Epsilons and Deltas is to perform menial labor and tasks that require little intelligence or creativity. They are essentially slaves to the higher castes and are kept in a state of perpetual contentment through conditioning.

4. Can Epsilons and Deltas ever achieve higher status in society?

In the novel, it is nearly impossible for Epsilons and Deltas to move up in society due to their conditioning and lack of opportunities. However, there are rare cases where some individuals break free from their conditioning and challenge the status quo.

5. How do Epsilons and Deltas differ from Alphas and Betas?

Epsilons and Deltas are physically and mentally inferior to Alphas and Betas. They are also conditioned to have less desire for material possessions and are content with their lowly status in society. Alphas and Betas, on the other hand, are the ruling class and have more individuality and freedom.

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