Comparison Theorem and Limits of integration

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The discussion centers on the limitations of the comparison theorem in integration, specifically regarding the necessity of having limits from a constant value to infinity rather than from negative infinity to infinity. Participants highlight that the comparison theorem is applicable when evaluating improper integrals, particularly when one function can be compared to another that is easier to integrate. An example provided illustrates that for x ≥ 1, the function x/(1+x^2) can be compared to 1/(2x), which aids in determining convergence. The conversation emphasizes the importance of establishing proper bounds for the comparison theorem to ensure valid conclusions about integrals. Understanding these constraints is crucial for correctly applying the theorem in calculus.
Painguy
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Homework Statement



Why is it that when using the comparison theorem my limits of integration must be from a constant value to infinity and not from negative infinity to infinity?

For example ∫ x/(1+x^2) dx from -∞ to ∞
 
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Can you be more specific about what comparison theorem you are talking about?

For your example, notice that if ##x \geq 1##, you can easily check that
$$\frac{x}{1+x^2} \geq \frac{1}{2x}$$
What does that tell you?
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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