Epsilon-Delta Proof of limit approaching infinity

In summary, the conversation is about proving the limit of a function as x approaches infinity. After providing their attempt at a solution and receiving feedback, the person concludes that their proof is sufficient.
  • #1
RPierre
10
0
**DISCLAIMER - I am super bad at LaTeX**

Homework Statement



Prove

[tex]\lim_{x \rightarrow \infty}\frac{1}{1+x^2} = 0[/tex]


Homework Equations



I Think I proved it, but I feel like I'm missing something to make this a proof of ALL [tex]\epsilon[/tex]>0 and not just one case. Maybe I did it right. I really don't know. Just looking for a second opinion and/or advice on [tex]\epsilon[/tex]-[tex]\delta[/tex] proofs.

3.Attempt at a Solution

the definition logically is if [tex]x > N[/tex] then [tex]|f(x) - L| > \epsilon[/tex] for some [tex]N,\epsilon > 0[/tex]

Setting [tex]N=\sqrt{\frac{1-\epsilon}{\epsilon}}[/tex]

x > N

[tex]\Rightarrow[/tex] [tex] x > \sqrt{\frac{1-\epsilon}{\epsilon}}[/tex]

[tex]\Rightarrow[/tex] [tex] x^2 > \frac{1-\epsilon}{\epsilon}[/tex]

[tex]\Rightarrow[/tex] [tex] x^2 + 1 > \frac{1}{\epsilon}[/tex]

[tex]\Rightarrow[/tex] [tex] \frac{1}{1+x^2} < \epsilon [/tex]

[tex] \frac{1}{1+x^2} = f(x) [/tex]

and since N > 0, and x > N, it is implied x > 0 and therefore [tex]|f(x)| = f(x)[/tex]

I'm not sure if this is a good enough proof? Thanks in Advance :)
 
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  • #2
Yes, your argument is fine. I would start it and end it slightly differently:

Begin with:

Suppose ε > 0 Then your next line

Let N = ... is OK
Then, if x > N...
...
...
[tex]
\frac{1}{1+x^2} < \epsilon
[/tex]

So

[tex]
|\frac{1}{1+x^2}| < \epsilon
[/tex]

And stop there.
 

1. What is the Epsilon-Delta Proof of limit approaching infinity?

The Epsilon-Delta Proof is a method used to rigorously prove that a function has a limit approaching infinity. It involves choosing a small value (epsilon) and showing that for any input greater than a certain value (delta), the output of the function will be within epsilon of the desired limit.

2. Why is the Epsilon-Delta Proof important?

The Epsilon-Delta Proof is important because it provides a precise and rigorous way to prove the existence of a limit approaching infinity. This is useful in many areas of mathematics and science, as well as in engineering and technology.

3. How do you use the Epsilon-Delta Proof to prove a limit approaching infinity?

To use the Epsilon-Delta Proof, you first choose a small value (epsilon) and then find a value (delta) such that for any input greater than delta, the output of the function will be within epsilon of the desired limit. This can be done by manipulating the definition of a limit and using algebraic techniques.

4. Are there any limitations to using the Epsilon-Delta Proof?

While the Epsilon-Delta Proof is a powerful tool for proving limits approaching infinity, it can be quite challenging and time-consuming to use. It also requires a good understanding of algebra and limits, so it may not be suitable for beginners.

5. Can the Epsilon-Delta Proof be used for any function?

The Epsilon-Delta Proof can be used for any function, as long as the function meets the criteria for having a limit approaching infinity. This includes functions that are continuous, differentiable, and have a finite slope at the point in question.

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