[Question] Limit and Integration

[Curl]

Suppose f(x,y) is some continuous, smooth function with the property
$$\lim_{y \to 0}f(x,y)=1$$
and g(x) is some other continuous smooth function.
I want to know if this is true:
$$\lim_{y \to 0} \int_{a}^{b}g(x)f(x,y)dx =? \int_{a}^{b}g(x)dx$$

How can I show that it is or isn't true? For which case will it be true or not true?

Thanks

 

[micromass]

Check out Lebesgue's monotone convergence theorem: http://en.wikipedia.org/wiki/Monotone_convergence_theorem (in the middle of the wiki page)

and his dominated convergence theorem: http://en.wikipedia.org/wiki/Dominated_convergence_theorem

 

[Curl]

Oh this is actually easy to see if you think of the integral as a Riemann sum.
I don't know why I always think of the solution right after I post the question.

So this is true in general, correct?

 

[micromass]

No, it certainly is not true in general. And I don't really know how Riemann sums help you here.

The problem is that you want to interchange two limits, this is not always allowed.

 

[CellCoree]

for your question! I would approach this problem by first understanding the definitions of limit and integration. A limit is the value that a function approaches as the input approaches a certain value, and integration is the process of finding the area under a curve.

In this case, we are looking at the limit of an integral. The integral is essentially the sum of infinitely small rectangles under a curve, and the limit is the value that these rectangles approach as the width of the rectangles approaches zero.

Based on the given information, we can see that the limit of the function f(x,y) as y approaches 0 is 1. This means that as y gets closer and closer to 0, the value of f(x,y) gets closer and closer to 1.

Now, let's consider the limit of the integral \int_{a}^{b}g(x)f(x,y)dx as y approaches 0. As y approaches 0, the function f(x,y) approaches 1, and the integral becomes \int_{a}^{b}g(x)dx. This is because the value of f(x,y) becomes constant (1) and can be factored out of the integral. Therefore, the limit of the integral is equal to the integral without the limit, or in other words, the limit and the integral can be interchanged.

To show that this is true, we can use the definition of the limit and the properties of integrals. We can also use the squeeze theorem, where we find two functions that are always smaller and larger than the original function, and show that they have the same limit.

In terms of when this will be true or not true, it will depend on the specific functions f(x,y) and g(x). As long as the conditions for the interchange of limit and integral hold (such as continuity and smoothness), then the statement will be true. However, if these conditions are not met, then the statement may not hold. It is important to carefully consider the properties of the functions involved to determine if the statement is true or not.