# [Question] Limit and Integration

1. Feb 22, 2012

### 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

2. Feb 22, 2012

### micromass

Staff Emeritus
3. Feb 22, 2012

### Curl

Oh this is actually easy to see if you think of the integral as a Riemann sum.
I dont know why I always think of the solution right after I post the question.

So this is true in general, correct?

4. Feb 23, 2012

### micromass

Staff Emeritus
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.