Switching Limit & Integral in Limits & Integrals

In summary, when taking the limit of an integral, it is possible to switch the composition as long as there is no infinity or indeterminate expression involved. This can be done if both the limit and integral are "uniformly convergent".
  • #1
moo5003
207
0
If you are taking the limit of an integral, can you switch the composition ie: take the integral of a limit if the limit and integral are on separate variables?

Ie:

Lim of z to a [integral over alpha [f(x)/((x-z)(x-a)^2) dx]

=

Integral over alpha[Lim of z to a[f(x)/((x-z)(x-a)^2) dx]

=

Integral over alpha[f(x)/(x-a)^3 dx]
 
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  • #2
Since an integration involves limits, and switching the order of taking limits is a tricky business, i'd say that you could do the permutation, as long as there's no infinity (or no indeterminate expression under the limit sign) involved...My guess...
 
  • #3
Generally speaking, if both limit and integral are "uniformly convergent" then they can be interchanged.
 

FAQ: Switching Limit & Integral in Limits & Integrals

What is the purpose of a switching limit in limits and integrals?

A switching limit is used to define the boundary or endpoint of a function or integral. It allows us to determine the behavior of a function or integral as it approaches a certain value or point.

How is a switching limit different from a traditional limit?

A traditional limit is a constant value that a function approaches as the independent variable approaches a specific value. A switching limit, on the other hand, is a variable that can change depending on the function and the value it is approaching.

How are switching limits used in integrals?

In integrals, switching limits are used to define the interval over which the function is being integrated. They indicate the lower and upper bounds of the integral and allow us to calculate the total area under the curve of the function.

What is the purpose of an integral in calculus?

An integral is used in calculus to find the area under a curve or the accumulation of a function over a given interval. It is also used to solve various real-world problems, such as finding the distance traveled by an object or the volume of a three-dimensional shape.

Can switching limits be applied to any type of function?

Yes, switching limits can be applied to any type of function, including polynomial, exponential, trigonometric, and logarithmic functions. They are a fundamental concept in calculus and are used to evaluate and analyze a wide range of functions.

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