Lim fn(x) = f(x) but lim ∫ |f(x)-fn(x)|dx ≠ 0 ?

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In summary, the conversation discussed finding an example of a sequence of integrable functions that satisfies lim fn(x) = f(x) as n -> ∞, but lim ∫ |f(x)-fn(x)|dx ≠ 0 (as n -> ∞). One suggested function, fn = (x + x/n), was determined to not work due to the limit of the integrals being 0. Another suggested function, fn(x) = |1/n| for x ∈ [0, n] and fn(x) = 0 elsewhere in [-infinity, infinity], was determined to work as it converges to 0 uniformly and the integral is 1 for all n in N. The term "integrable" was discussed
  • #1
pulin816
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0

Homework Statement



Hey, I have another questions,
I need to find an example of a sequence of integrable functions fn:R -> R, n =1, 2, ...
such that

lim fn(x) = f(x) (as n -> ∞)
but lim ∫ |f(x)-fn(x)|dx ≠ 0 (as n -> ∞)​
(with integral from - to + infinity)

The Attempt at a Solution



I've tried
fn = (x + x/n)
and f = x

the first conditions would be satisfied, but on the other hand,
will the limits and the integral be interchangeable? I've read that it is only permitted if the expression inside is bounded. |x/n| can't be bounded since it has an absolute sign wrapped around or would it?

Any suggestions? Thank you !

p.s. Would the term 'integrable' here mean a function that is reinmann integrable?
 
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  • #2
I wouldn't worry about the interchange of limits, at least not right now. The function you suggested would not work because the limit of the integrals is 0 anyway.
 
  • #3
Thanx LeonhardEuler:

I tried coming up with a nother function, what about

fn(x) = |1/n| for x ∈ [0, n] and
fn(x) = 0 elsewhere in [-infinity, infinity]​
fn(x) will converge to 0 function uniformly
however, while ∫ |fn(x)| dx = 2 for all n in N

?
 
  • #4
Yeah, that works.
 
  • #5
Except the integral is 1, not 2.
 
  • #6
oh yeah, my mistake.
Thanks!
 

1. What does it mean for a function's limit to equal the function itself?

In mathematics, a function's limit is the value that the function approaches as the input variable gets closer and closer to a particular value. If the limit of a function at a certain value is equal to the function itself, it means that the function is continuous at that point. This also means that the function is well-behaved and does not have any abrupt changes or discontinuities at that point.

2. What is the significance of lim ∫ |f(x)-fn(x)|dx ≠ 0 ?

This statement means that the limit of the integral of the absolute value of the difference between the function and its approximation is not equal to zero. This indicates that the approximation is not accurate enough and there is a significant difference between the two functions. This could be due to various reasons such as the approximation being too coarse or the function having sharp changes or discontinuities.

3. Can a function's limit be equal to the function itself but the limit of the integral not equal to zero?

Yes, it is possible for a function's limit to be equal to the function itself but the limit of the integral not equal to zero. This can happen when the function has a jump discontinuity or a removable discontinuity. In this case, the function is still continuous at that point, but the integral of the absolute value of the difference between the function and its approximation may not approach zero.

4. How does the size of the interval of integration affect the limit of the integral?

The size of the interval of integration can greatly affect the limit of the integral. If the interval is too large, the approximation may not be accurate enough and the limit of the integral may not approach zero. However, as the size of the interval decreases, the limit of the integral may approach zero and the approximation becomes more accurate. This is why it is important to carefully choose the interval of integration when approximating a function.

5. What are some possible solutions if lim ∫ |f(x)-fn(x)|dx ≠ 0 ?

If the limit of the integral is not equal to zero, it means that the approximation is not accurate enough. Some possible solutions to improve the accuracy of the approximation could be to decrease the interval of integration, increase the number of subintervals, or use a more precise method of approximation such as Simpson's rule or the Trapezoidal rule. Additionally, if the function has sharp changes or discontinuities, it may be necessary to use a different approximation method that can handle such features.

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