Limit of (x^n + y^n)^(m/n) n->infinity Proof

In summary, the conversation discusses a proof for the equation \lim_{n\to \infty} (x^n + y^n)^{\frac{m}{n}} = (max\{x,y\})^m\;\;\;\forall x,y > 0, using the concept of limits and the fact that \lim_{x\to x_0} (f(x))^n = (\lim_{x\to x_0} f(x))^n. The proof is presented in a detailed and clear manner, explaining each step and considering all cases, resulting in a convincing and textbook-like proof. The importance of providing detailed explanations in proof-writing is also mentioned.
  • #1
BrianMath
26
0

Homework Statement


Show that
[tex]\lim_{n\to \infty} (x^n + y^n)^{\frac{m}{n}} = (max\{x,y\})^m\;\;\;\forall x,y > 0[/tex]
where [itex]max\{x,y\}[/itex] outputs the greater of the two.


Homework Equations


[tex]\lim_{x\to x_0} (f(x))^n = (\lim_{x\to x_0} f(x))^n[/tex]


The Attempt at a Solution


It's attached as a pdf file. I think I have the proof right, I would just like to know your opinions on how it is laid out. Is it too wordy, or too informal? I'm still not used to writing proofs in a way to convey information to other people, and I've heard that the only way to pick it up is through practice, so that's what I'm doing. :biggrin:
 

Attachments

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  • #2
well, your proof is correct and convincing, and I don't think that it is too informal, you've just explained every step carefully in details which is good, that doesn't make it wordy or informal. you can avoid going into such details in your exam paper. you have to consider the case when x=0 separately if you want it to be more rigorous. (you're not allowed to factor out x^n if it's zero)
 
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  • #3
You're missing the case where x=y :smile:
 
  • #4
micromass said:
You're missing the case where x=y :smile:

Ah, yes, thank you. :smile:
I've taken care of that, how is it now? :biggrin:


The way I like to write my proofs is to explain every step in detail so that even someone with a basic knowledge of calculus could understand. I think it's mostly to convince myself that I know what I'm doing, but I also want to make sure when someone grades what I do that I don't get points taken away for not providing enough steps or justification.:-p
 

Attachments

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  • #5
BrianMath said:
Ah, yes, thank you. :smile:
I've taken care of that, how is it now? :biggrin:

Now it's all fine!

The way I like to write my proofs is to explain every step in detail so that even someone with a basic knowledge of calculus could understand. I think it's mostly to convince myself that I know what I'm doing, but I also want to make sure when someone grades what I do that I don't get points taken away for not providing enough steps or justification.:-p

I think your proof was quite beautiful. If they showed it to me, then I would say that it's from a textbook. I like how you explained what you're doing, instead of just writing the calculation...
Keep writing proofs like this!
 
  • #6
micromass said:
Now it's all fine!
I think your proof was quite beautiful. If they showed it to me, then I would say that it's from a textbook. I like how you explained what you're doing, instead of just writing the calculation...
Keep writing proofs like this!

Thank you for the compliment! I only ever see proofs that way in textbooks, so I was worried that I might be writing a bit too textbook-stylish (but now I see that's a good thing). I'm self taught for the most part (since I was 13), so I know when I do computations right, but the kind of ambiguity involved in the art of proof writing always made me wary of whatever I write.

Now I'm confident in my proof-writing abilities. :smile:
 

Related to Limit of (x^n + y^n)^(m/n) n->infinity Proof

1. What is the purpose of finding the limit of (x^n + y^n)^(m/n) as n approaches infinity?

The purpose of finding this limit is to determine the behavior of the expression (x^n + y^n)^(m/n) as n gets larger and larger. This can help us understand the long-term trends and patterns of the expression, which can be useful in various mathematical and scientific applications.

2. How do you approach this type of limit proof?

To prove the limit of (x^n + y^n)^(m/n) as n approaches infinity, we can use the concept of L'Hôpital's rule, which states that for certain types of indeterminate forms (such as 0/0 or ∞/∞), the limit can be found by taking the derivative of the numerator and denominator and then evaluating the limit again.

3. Can you provide an example of using L'Hôpital's rule to prove this limit?

Sure, let's take the limit of (x^n + y^n)^(m/n) as n approaches infinity. We can rewrite this expression as (x^n + y^n)^(1/n)^(m), which can be simplified to (x + y)^(m). Now, we can use L'Hôpital's rule by taking the derivative of the numerator and denominator, which gives us m(x + y)^(m-1) / (x + y)^(m-1). We can then evaluate the limit again, which gives us (x + y)^(m-1), a finite number. Therefore, the limit of (x^n + y^n)^(m/n) as n approaches infinity is (x + y)^(m-1).

4. Are there any other methods for proving this type of limit?

Yes, another method that can be used to prove this limit is the squeeze theorem. This theorem states that if a function is "sandwiched" between two other functions that have the same limit, then the middle function also has the same limit. In this case, we can find two functions that are always greater and less than (x^n + y^n)^(m/n), and prove that they both have the same limit as n approaches infinity.

5. What are some real-life applications of this type of limit proof?

Finding the limit of (x^n + y^n)^(m/n) as n approaches infinity can be useful in various fields such as physics, engineering, and economics. For example, it can be used to analyze the behavior of a population over time, the growth rate of a disease, or the decay rate of a radioactive substance. It can also be applied in financial forecasting and predicting trends in the stock market.

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