Delta-Epsilon Proof: Prove lim_{x\implies 1} \frac{2}{x-3} = -1

In summary, the conversation discusses how to prove a limit using delta-epsilon and finding a suitable value for delta. The proof strategy is to show that |(2/(x-3)) + 1| < epsilon, restricting delta to be a function of epsilon alone. The conversation goes on to determine a suitable value for delta by setting |x-1| < 1 ≤ delta, and then finding a constant K that satisfies the inequality. The conversation concludes by mentioning the need to prove that the chosen value for delta works.
  • #1
knowLittle
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3

Homework Statement


Prove that
## lim_{x\implies 1} \frac{2}{x-3} = -1 ##

Use delta-epsilon.

The Attempt at a Solution


Proof strategy:
## | { \frac{ 2}{x-3} +1 } | < \epsilon #### \frac{x-1}{x-3} < \epsilon ##
, since delta have to be a function of epsilon alone and not include x. I need to restrict delta
## |x-1 | < 1 \leq \delta \\ -3 < x-3 < -1 ##

I know that there's something wrong. Help?

What if I say that
## -2 = x -3 \\ 1 =x \\ \frac{ 1-1}{-2} < \epsilon \\ \delta=min(1, \epsilon)##

Does it make any sense?
 
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  • #2
knowLittle said:

Homework Statement


Prove that
## lim_{x\implies 1} \frac{2}{x-3} = -1 ##

Use delta-epsilon.

The Attempt at a Solution


Proof strategy:
## | { \frac{ 2}{x-3} +1 } | < \epsilon #### \frac{x-1}{x-3} < \epsilon ##
, since delta have to be a function of epsilon alone and not include x. I need to restrict delta
## |x-1 | < 1 \leq \delta \\ -3 < x-3 < -1 ##

I know that there's something wrong. Help?

You don't know how small [itex]\delta[/itex] may need to be, but you can decide that it's not going to be bigger than 1. That gives you [tex]|x - 1| < \delta \leq 1.[/tex] Now you need [tex]
\left| \frac{x - 1}{x - 3 } \right| < \frac{\delta}{|x - 3|} < \epsilon.[/tex] You can ensure that by finding a constant [itex]K > 0[/itex] such that [tex]\frac{\delta}{|x - 3|} < K\delta[/tex] when [itex]|x - 1| < \delta \leq 1[/itex], and insisting that [itex]K\delta < \epsilon[/itex].
 
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  • #3
You're almost there, now prove that your choice for delta works.
 

1. What is a delta-epsilon proof?

A delta-epsilon proof is a method used in mathematics to formally prove the limit of a function. It involves choosing a small interval (delta) around the limit point and showing that for any value within that interval, there exists another interval (epsilon) around the limit point where the function's output is within a desired range.

2. How do you use a delta-epsilon proof to prove a limit?

To prove a limit using a delta-epsilon proof, you must first define the small interval (delta) around the limit point and then determine the desired range of outputs (epsilon). Then, you must show that for any value within the delta interval, there exists an epsilon interval around the limit point where the function's output falls within the desired range.

3. Why is the delta-epsilon proof important in mathematics?

The delta-epsilon proof is important in mathematics because it provides a rigorous and formal way to prove limits of functions. It allows mathematicians to make precise and accurate statements about the behavior of a function near a specific point.

4. Can you provide an example of a delta-epsilon proof?

Sure, an example of a delta-epsilon proof for the limit lim_{x\implies 1} \frac{2}{x-3} = -1 would involve choosing a small delta interval around the limit point of 1, such as 0.5. Then, we must determine the desired range of outputs (epsilon), which in this case would be 0.5. Finally, we must show that for any value within the delta interval of 0.5, there exists an epsilon interval around the limit point of 1 where the function's output is within the range of 0.5. This can be done by manipulating the function algebraically and showing that it satisfies the definition of a limit.

5. What are the key components of a delta-epsilon proof?

The key components of a delta-epsilon proof are the small interval (delta) around the limit point, the desired range of outputs (epsilon), and the proof that for any value within the delta interval, there exists an epsilon interval around the limit point where the function's output falls within the desired range. Additionally, the algebraic manipulation of the function and use of the definition of a limit are important components of a delta-epsilon proof.

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