Why is it fine to assume .5 | x - 4 | < e in an epsilon-delta proof?

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In summary, The conversation discusses a question regarding an epsilon-delta proof. The proof involves an inequality that shows if \left|\ x-4\right|\ < \epsilon, then \left|\sqrt{x}-2\right|\ < \epsilon. The last line of the proof seems peculiar because it is necessary for the proof, but it can be derived in reverse from the conclusion to the hypothesis.
  • #1
bjgawp
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Hey there everyone. I was looking at an epsilon-delta proof I did and realized that I wasn't exactly sure why one of my statements was true:

http://img255.imageshack.us/img255/7356/proofmy5.jpg

On the third last line, why is it fine to assume that .5 | x - 4 | < e is true? Isn't there a chance that .5 | x - 4 | may be greater than e?

Thanks in advance!

Edit: Hmm, wonder why the IMG tags don't work ...
 
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  • #2
it's not okay to assume that.The inequality [tex]\left|\sqrt{x}-2\right|\leq \frac{1}{2} \left|\ x - 4 \right|\ [/tex] shows that if you restrict the x-values so that [tex] \left|\ x - 4\right|\ <2\epsilon [/tex] for the given [tex]\epsilon[/tex], then you get

[tex]\left|\sqrt{x}-2\right|\leq \frac{1}{2} \left|\ x - 4 \right|\ < \frac{1}{2} (2\epsilon) =\epsilon[/tex]

Also, the last line, 'And since . . .' is a bit weird. Remember that you're showing that such a delta exists in the first place.
 
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  • #3
How does [tex]\left|\sqrt{x}-2\right|\leq \frac{1}{2} \left|\ x - 4 \right|\ [/tex] show that [tex] \left|\ x - 4\right|\ <2\epsilon [/tex] without assuming [tex]\frac{1}{2} \left|\ x - 4 \right|\ < \epsilon [/tex] first? Sorry. I don't see how the first inequality connects with it being less than epsilon. Thanks for the help!

Oh and yeah, I typed this up haphazardly without thinking what I meant by the last line. Thanks again!
 
  • #4
remember that you fixed [tex]\epsilon[/tex] as a positive number.

the inequality shows that IF you make [tex]\left|\ x - 4\right|\ < \epsilon[/tex] , THEN you get [tex]\left|\sqrt{x}-2\right|\<\epsilon[/tex].

Read over the definition of a limit carefully and see how it applies to this particular problem:

We say that [tex]\lim_{x\to\\a}f(x)=v [/tex] whenever,

for all [tex] \epsilon > 0 [/tex], there is some [tex]\delta >0[/tex] such that,

whenever we have [tex]0<\left|\ x-a \right|\ < \delta[/tex], it follows that [tex]\left|\ f(x) - v\right|\ < \epsilon [/tex]
 
  • #5
The reason that last line seems a bit "peculiar" is that it is really a peculiarity of the way we do proofs of limits.

The definition of limit requires that we show that, for a specific [itex]\delta[/itex], if [itex]|x-a|< \delta[/itex], then [itex]|f(x)- L|< \epsilon[/itex]. But what we do is work the other way- assuming a value of [itex]\epsilon[/itex], we calculate the necessary [itex]\delta[/itex]. The point is that every step is "reversible"- you could start from, in this case, |x-4|< [itex]\delta[/itex] and, by just going through the derivation "in reverse" arrive at [itex]|\sqrt{x}- 2|< \epsilon[/itex].

That's sometimes referred to as "synthetic" proof. Again, we go from the conclusion we want to the hypothesis- but it is crucial that every step be "reversible"!
 

FAQ: Why is it fine to assume .5 | x - 4 | < e in an epsilon-delta proof?

What is an Epsilon-delta Proof?

An Epsilon-delta Proof is a type of mathematical proof used to rigorously prove the limit of a function. It is commonly used in calculus and analysis to show that a function approaches a certain value as its input approaches a specific value.

How does an Epsilon-delta Proof work?

An Epsilon-delta Proof involves choosing a small positive number, epsilon, and showing that for any input value, there exists a corresponding delta value that ensures the output of the function will be within epsilon of the desired limit. This is done by manipulating the function and using algebraic and logical reasoning.

Why are Epsilon-delta Proofs important?

Epsilon-delta Proofs are important because they provide a rigorous and logical way to prove the limit of a function. They help to ensure the validity of mathematical statements and provide a deeper understanding of the properties of functions and limits.

What are some common challenges in constructing an Epsilon-delta Proof?

One common challenge in constructing an Epsilon-delta Proof is choosing the appropriate epsilon and delta values. These values must be carefully selected to ensure that the desired limit is achieved without making the proof too complex. Another challenge is properly manipulating the function to show the relationship between epsilon and delta.

How can one improve at constructing Epsilon-delta Proofs?

The best way to improve at constructing Epsilon-delta Proofs is through practice. It is important to fully understand the concepts behind the proof and to practice using them in different scenarios. It may also be helpful to seek guidance from a teacher or tutor to get feedback and tips for improvement.

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