Delta - epsilon proof

In summary, the conversation discusses a proof for the limit of a function and how to relate ε to δ in order to prove the limit exists. The conversation includes a trial and error approach and ultimately concludes that δ should be the smaller of 3ε and 1.
  • #1
gimpy
28
0
Ok well i did problems like this before but now I am having trouble with this one for some reason.

Let [tex]f(x) = \frac{1}{\sqrt{x}}[/tex]. Give a [tex]\delta[/tex] - [tex]\epsilon[/tex] proof that [tex]f(x)[/tex] has a limit as [tex]x \rightarrow 4[/tex].

So the defn of a limit is
[tex]\forall \epsilon > 0 \exists \delta > 0[/tex] such that whenever [tex]0 < |x - 4| < \delta[/tex] then [tex]|f(x) - l| < \epsilon[/tex]
Assuming the limit we are trying to prove is [tex]l[/tex].

So i know i somehow have to turn [tex]|f(x) - l| < \epsilon[/tex] into something with [tex]x - 4 < ...[/tex] and that will prove that the limit exists. Am i correct? Am i on the right track? can i assume that [tex]l = \frac{1}{2}[/tex] since [tex]\frac{1}{\sqrt{4}}[/tex]?
 
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  • #2
1. There exists no general technique to find a limit.
What the d,e-business is about in this case (I think), is to investigate whether an arbitrarily chosen number l is, in fact, the limit value.
So, it is perfectly legitimate to investigate whether the number 1/2 is the limit value or not.
 
  • #3
Well, I tried to do this, but I sort of got stuck at the end, so if anyone can check & let us know where I've gone wrong and how to complete it correctly, I'd appreciate it.

I started with the "guess" that the limit is 1/2, and therefore we need to show that
[tex] \left| \frac{1}{\sqrt{x}} - \frac{1}{2} \right| < \epsilon \: \text{whenever}\: 0 < |x-4| < \delta[/tex]
If I combine the fractions and require δ < 1, then 3 < x < 5 and √x > 1, so
[tex] \left| \frac{1}{\sqrt{x}} - \frac{1}{2} \right| = \left| \frac{2 - \sqrt{x}}{2\sqrt(x)}\right| < \left| \frac{2 - \sqrt{x}}{2}\right| < \epsilon [/tex]

[tex] -\epsilon \;< \; \frac{2 - \sqrt{x}}{2} \;<\; \epsilon [/tex]

[tex] -2\epsilon \;< \; 2 - \sqrt{x} \;<\; 2\epsilon [/tex]

[tex] -2\epsilon -2 \;<\; - \sqrt{x} \;<\; 2\epsilon -2 [/tex]

[tex] 4\epsilon^2 +8\epsilon +4 \;>\; x \;>\; 4 - 8\epsilon + 4\epsilon^2[/tex]

[tex] 4\epsilon^2 + 8\epsilon \;>\; x-4\; >\; 4\epsilon^2 - 8\epsilon[/tex]

[tex] 4\epsilon^2 - 8\epsilon \text{ is no good, since for small values of }\epsilon \text { that will be negative,}[/tex]
[tex]\text{ so does this mean that for any value of }\epsilon \text{ , choosing}[/tex]
[tex] \delta \;<\; 4\epsilon^2 + 8\epsilon \:\text{satisfies the proof?}[/tex]

I really don't feel good about this conclusion and I'm getting confused over the directions of the inequalities, so someone please help.

Edit: well, the more I look at it, the more convinced I am that this is wrong, so if anyone can help, please do.
 
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  • #4
|2- sqrt(x)| < (2+sqrt(x))(2+sqrt(x)) = |4-x| < d

that any help?
 
  • #5
No, I still don't see it.
I assume you meant to write |2- sqrt(x)| < (2-sqrt(x))(2+sqrt(x)) = |4-x| < d

But we need
[tex] \left| \frac{2 - \sqrt{x}}{2\sqrt{x}}\right| < \epsilon[/tex]

whenever
[tex] \left|2-\sqrt{x}\right| \left|2+\sqrt{x}\right| = \left|4-x\right| < \delta [/tex]

and I'm still not seeing how to relate ε to δ

Presumably we need
[tex]\delta = ({2+\sqrt{x})\epsilon}[/tex]
but I don't know how to accomplish that.
 
  • #6
I bet you know something smaller than (2 + &radic;x)&epsilon; that doesn't depend on x!
 
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  • #7
I should probably stick to my own homework. :redface:

That was all wrong. Still working on it.
 
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  • #8
OK, now I let δ = 3ε, and keep the restriction 3<x<5 which is reasonable since we're interested in values of x close to 4.

So if
[tex] 0 < |x-4| < \delta[/tex]
then
[tex] 0 < |\sqrt{x}-2|\times(\sqrt{x}+2) < 3\epsilon[/tex]
Now divide by 3
[tex] 0 < |\sqrt{x}-2|\times \frac{(\sqrt{x}+2)}{3} < \epsilon[/tex]
And since √x is definitely bigger than 1, we know that
[tex] \frac{(\sqrt{x}+2)}{3} > 1[/tex]
so
[tex] \left| \frac{1}{\sqrt{x}} - \frac{1}{2} \right| = \left| \frac{2 - \sqrt{x}}{2\sqrt{x}}\right| < \left|{2 - \sqrt{x}}\right| < |\sqrt{x} - 2|\times \frac{(\sqrt{x}+2)}{3} < \epsilon [/tex]

Is that it? Did I spend all this time just to find that? :eek:

Now it looks right to me. Is anything missing?

Is there some obvious clue that I should have seen that would have told me to try δ = 3ε?

Thanks for your help, Matt & Hurkyl. Please let me know if this still needs more work.

(Edited to correct typos.)
 
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  • #9
Aside from the typo, looks right.

Incidentally, if you wanted to sew up the loose end, the trick to saying:

"OK, now I let &delta; = 3&epsilon;, and keep the restriction 3<x<5 which is reasonable since we're interested in values of x close to 4."

is "let &delta; be the smaller of 3&epsilon; and 1"

So that way you have both |x-4| < 3&epsilon; and |x-4| < 1
 

1. What is a delta-epsilon proof?

A delta-epsilon proof is a mathematical technique used to formally prove the limit of a function. It involves using a precise definition of a limit, where the difference between the input and the limit is represented by epsilon and the distance between the input and the limit is represented by delta.

2. Why is a delta-epsilon proof important?

A delta-epsilon proof is important because it provides a rigorous and logical way to prove limits in mathematics. It ensures that the limit is true for all values of epsilon and delta, making it a more reliable method than other techniques.

3. How does a delta-epsilon proof work?

A delta-epsilon proof works by showing that for any given epsilon value, there exists a corresponding delta value that satisfies the definition of a limit. This is done by manipulating the function and setting up an inequality that can be solved for delta in terms of epsilon.

4. Can a delta-epsilon proof be used for all functions?

Yes, a delta-epsilon proof can be used for any function that has a limit. However, it may be more complicated for some functions and may require more advanced mathematical techniques.

5. What are some common mistakes to avoid in a delta-epsilon proof?

Some common mistakes to avoid in a delta-epsilon proof include using the wrong definition of a limit, not considering all values of epsilon and delta, and not showing all the steps in the proof. It is important to be precise and thorough when using this technique to ensure a correct proof.

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