How Do You Prove a Limit Using the ε-δ Definition?

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In summary, the conversation is about using the ε-δ definition of the limit to prove that for any ε>0, there exists a δ>0 such that for any x with 0 < |x - 3| < δ, |(2x - 6)/(x-3) - 2|< ε. The conversation discusses simplifying the expression and finding the value of δ that would satisfy the limit. It is mentioned that the simplified form of the expression is 0 when (x-3) ≠ 0, but it is still necessary to show that for any x with 0 < |x - 3| < δ, the expression is less than ε.
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SithsNGiggles
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Homework Statement



Use the ε-δ definition of the limit to prove that

lim as x -> 3 of (2x - 6)/(x-3) = 2

Homework Equations



The Attempt at a Solution



I've started a preliminary analysis for the proof:
For any ε>0, find δ=δ(ε) such that 0 < |x - 3| < δ implies that |(2x - 6)/(x-3) - 2|< ε.
Simplifying:
|(2x - 6)/(x-3) - 2| = |(2x - 6 - 2(x - 3))/(x-3)| = |(2x - 6 - 2x + 6)/(x - 3)| = 0

Where do I go from here? Have I just proven that since the simplified form is 0, which is less than ε, and therefore proven the limit?
 
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  • #2
What do you get when you reduce (2x-6)/(x-3) to lowest terms?
 
  • #3
2. Does that mean I've proved the limit? No formal proof required?
 
  • #4
No. Any δ > 0 will work, but you still need to show that for any x such that 0 < |x - 3| [STRIKE]< δ[/STRIKE] , |(2x - 6)/(x-3) - 2|< ε .

I crossed out the < δ side, because as long as (x-3) ≠ 0, (2x - 6)/(x-3) - 2 = 0 which, of course is less than ε .
 

FAQ: How Do You Prove a Limit Using the ε-δ Definition?

1. What is an ε-δ limit proof?

An ε-δ limit proof is a mathematical technique used to prove the convergence of a sequence or a function. It involves showing that for any given value of ε (epsilon), there exists a corresponding value of δ (delta) such that the distance between the function/sequence and its limit is less than ε.

2. Why is an ε-δ limit proof important?

An ε-δ limit proof is important because it provides a rigorous and formal way to prove the convergence of a sequence or function. It is also a fundamental concept in real analysis and is used to establish important theorems such as the continuity of a function.

3. What are the key steps in a ε-δ limit proof?

The key steps in an ε-δ limit proof are: 1) Setting up the problem and understanding the given limit statement, 2) Choosing a suitable value of δ based on ε, 3) Simplifying the expression to be less than or equal to ε, 4) Manipulating the expression to find an upper bound for δ, and 5) Writing a formal proof using the chosen value of δ.

4. How do I know if my ε-δ limit proof is correct?

You can check the correctness of your ε-δ limit proof by carefully reviewing each step and making sure that the chosen value of δ satisfies the conditions of the limit statement. You can also verify your proof with a peer or a tutor to ensure its accuracy.

5. Are there any tips for writing a successful ε-δ limit proof?

Some tips for writing a successful ε-δ limit proof include: 1) Carefully reading and understanding the given limit statement, 2) Choosing a value of δ that works for all values of ε, 3) Showing all steps and explanations clearly and logically, 4) Using proper mathematical notation and symbols, and 5) Checking for errors and making necessary corrections.

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