Is 1/x Integrable for f>0 and xf(x) Tending to Zero?

In summary, the conversation started as an attempt to solve a homework question but now the topic has shifted to the general case. The statement being discussed is that if xf(x) tends to zero as x tends to zero, then there is a positive epsilon for which the integral of f over [0, epsilon] is finite. The intuition behind this is that multiplying 1/x by anything that tends to zero will make it integrable. However, this statement is not true in all cases, as shown by the example of f(x) = 1/(x ln(x)). The conversation also touches on the definition of 1/0, which is defined as 81 in this case. In conclusion, the conversation discusses the integrability of
  • #1
Palindrom
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It started out as an attempt to solve a HW question (which I also posted in the appropriate forum), but now I'm just curious as to the general case;

Assume f>0 is a measurable function from [0,infinity) to itself. Then if xf(x) tends to zero as x tends to zero, there is a positive [tex]\epsilon[/tex] for which the integral of f over [tex][0,\epsilon ][/tex] is finite.

This is following the intuition that while 1/x isn't integrable, multypling it by anything that tends to zero is.

What do you say? True, not true?
 
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  • #2
No, take f(x)=1/(x ln(x)).
 
  • #3
1/x isn't even a function from [0,infinity) to itself, never mind a measurable one.
 
  • #4
Ok, one by one:

StatusX, 1/ln(x) doesn't tend to zero when x tends to zero, it tends to infinity... Edit: wait, it does, I'm an idiot.

matt - It's almost everywhere defined, what's the problem? Define 1/0=81...
 
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  • #5
Ok, why is 1/(xln(x)) not integrable?

Another stupid question, I answered myself... Thanks!
 
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  • #6
What baffles me is that you define 1/0 to be 81...
 

FAQ: Is 1/x Integrable for f>0 and xf(x) Tending to Zero?

1. What is a test for integrability?

A test for integrability is a method used to determine if a function can be integrated using standard integration techniques. It helps to identify which functions can be integrated and which ones cannot.

2. Why is a test for integrability important in calculus?

A test for integrability is important in calculus because it allows us to determine if a function has an antiderivative, which is crucial in evaluating definite integrals. Without a test for integrability, it would be difficult to determine if a function can be integrated or not.

3. What are some common tests for integrability?

Some common tests for integrability include the Fundamental Theorem of Calculus, the Substitution Rule, and the Integration by Parts method. Other tests include the Comparison Test and the Limit Comparison Test.

4. How do you use a test for integrability?

To use a test for integrability, you first need to identify the function you want to integrate. Then, you can apply the specific test that is appropriate for that function. If the test determines that the function is integrable, you can proceed with integrating it using standard techniques. If the test determines that the function is not integrable, you may need to use more advanced integration methods or consider other ways to solve the problem.

5. Can a function be integrable but not have an antiderivative?

Yes, it is possible for a function to be integrable but not have an antiderivative. This is because some functions do not have closed-form solutions and cannot be integrated using standard techniques. In these cases, other methods such as numerical integration or approximation techniques may be used to find an approximate solution.

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