Bringing a limit inside of an integral

In summary, bringing a limit inside of an integral means evaluating the integral by taking the limit of the function as the variable approaches a certain value. This technique can make it easier to solve difficult integrals and can lead to more precise calculations and simplification of complex functions. To bring a limit inside of an integral, you need to identify the variable to take the limit of and use limit rules to evaluate it. However, there are limitations to this method, as it may not always give accurate results and may not work for certain types of integrals. Despite this, bringing a limit inside of an integral can still be utilized in real-world applications, such as in physics and engineering, to solve problems and make predictions.
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Random Variable
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Homework Statement



Under what conditions does [itex] \lim_{s \to 0^{+}} \int_{0}^{\infty} f(x) e^{-sx} \ dx = \int_{0}^{\infty} f(x) \ dx [/itex] ?


Homework Equations





The Attempt at a Solution



If justification is ever offered, it's that [itex] \int_{0}^{\infty} f(x) \ dx [/itex] converges. But I'm not certain that's enough justification. It was suggested to me to also show that [itex] \int_{0}^{\infty} f(x) e^{-sx} \ dx [/itex] converges absolutely.
 
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  • #2
However, while I know what absolute convergence is, I'm not sure how to show it and that's where I'm stuck.
 

FAQ: Bringing a limit inside of an integral

1. What does it mean to "bring a limit inside of an integral"?

Bringing a limit inside of an integral means to evaluate the integral by taking the limit of the function as the variable approaches a certain value, instead of integrating the entire function.

2. Why would someone want to bring a limit inside of an integral?

Bringing a limit inside of an integral allows for easier evaluation of integrals that are otherwise difficult or impossible to solve. It also allows for more precise calculations and can help to simplify complex functions.

3. How do you bring a limit inside of an integral?

To bring a limit inside of an integral, you first need to identify the variable in the integral that you want to take the limit of. Then, you can use the limit rules to evaluate the limit and substitute the resulting value into the integral.

4. What are the limitations of bringing a limit inside of an integral?

Bringing a limit inside of an integral may not always give an accurate result, especially if the limit is taken at a point where the function is undefined. It also may not work for certain types of integrals, such as those with discontinuities or infinite limits.

5. Can bringing a limit inside of an integral be used in real-world applications?

Yes, bringing a limit inside of an integral can be used in real-world applications, such as in physics and engineering, to calculate areas, volumes, and other quantities. It can also be used to model real-world situations and make predictions.

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