Limits - Epsilon Delta Proof for 2x^2-2<1

In summary, the conversation discusses solving for a limit and determining the correct value for delta in an epsilon-delta proof. The problem involves a limit as x approaches -1 and the limit is equal to 2x^2=2. The conversation also mentions using a delta value of 1/2(-2 + delta) to solve for the limit. There is also a discussion about the importance of including an epsilon value in an epsilon-delta proof.
  • #1
evry190
13
0
Okay so here's the limit lim(as x --> -1) 2x^2=2

I have |2x^2-2| < 1
2|x + 1||x - 1| < 1
|x + 1||x - 1| < 1/2

and 0 < |x + 1 | < delta
-1 - delta < x < -1 + delta
|x - 1| < -2 + delta

|x + 1| < 1/2(-2 + delta)

so is delta = 1/2(-2 + delta) correct? if so, how do i know which answer (since there will be 2) is right?
 
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  • #2
Hi evry190! :smile:

(have a delta: δ and an epsilon: ε :wink:)

Where's your ε?

(an epsilon delta proof has to have an epsilon …

the clue's in the name! :biggrin:)​

Start again, and try to prove that |2x^2-2| < ε. :smile:
 
  • #3
but in the problem the epsilon is 1
 
  • #4
evry190 said:
but in the problem the epsilon is 1

I don't understand … the problem was "the limit lim(as x --> -1) 2x^2=2" …

where's the ε = 1 in that? :confused:
 

1. What is a limit?

A limit is a fundamental concept in calculus that represents the value that a function approaches as the input variable gets closer and closer to a specific value. It is often denoted by the notation lim f(x).

2. What is an epsilon-delta proof?

An epsilon-delta proof is a method used to formally prove the existence of a limit for a given function. It involves choosing a small positive number (epsilon) and showing that for any input value (delta) that is within a certain distance from the limit, the output value of the function is within the range specified by epsilon.

3. How does the epsilon-delta proof work?

The epsilon-delta proof works by first choosing a small positive value for epsilon. Then, by manipulating the given inequality (2x^2 - 2 <1), we can find a corresponding value for delta that satisfies the conditions of the proof. This shows that for any input value within a certain distance from the limit, the output value will always be within the range specified by epsilon.

4. Why is the epsilon-delta proof important?

The epsilon-delta proof is important because it provides a rigorous and formal way of proving the existence of a limit. This proof is widely used in calculus and other branches of mathematics to prove fundamental concepts and theorems.

5. Can the epsilon-delta proof be used for any function?

Yes, the epsilon-delta proof can be used for any function, as long as the function satisfies the conditions for the proof. These conditions include continuity and differentiability, which are fundamental properties of most functions studied in calculus.

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