# Limit fo integral

1. Jun 21, 2011

### testito

hello guy ; i have a bizarre question in limit of integral , the question is :

determine :

$\lim_{n \to \infty } \int_{0}^{1} \frac{nf(x)}{n^{2} + x^{2}} dx$

i don't know really when start and when i finish , please tell me how do this and if you can give me some tutorial for this scope !!

2. Jun 21, 2011

### testito

i forget to tell you that f is continue on interval [0,1] !!

3. Jun 21, 2011

### Unit

The only idea I have is to argue that since f is continuous on a closed interval, then f is bounded. Perhaps you could use the Squeeze Theorem?

4. Jun 21, 2011

### Hootenanny

Staff Emeritus
Unit has it right. Once you realise that f(x) is bounded, the computation becomes trivial (no need for the squeeze theorem).

5. Jun 21, 2011

### Unit

I suggested squeeze because f(x) could be negative; saying A >= f and A --> L does not imply f --> L, unless f >= B and B --> L also.

6. Jun 21, 2011

### testito

can you determine the limit above exactly please !!
look the second question in this exercise is :

prove that f is Riemann integrable in [0,1] and :

$\int_{0}^{1} f(x) dx = \lim_{n \to \infty }\frac{1}{n}\sum_{k=0}^{n} f(\frac{k}{n})$

i'm really confused !!

7. Jun 22, 2011

### Hootenanny

Staff Emeritus
I agree that the squeeze theorem would be the way to formally proceed. However, it all depends on how rigerous one needs to be. Since the question only wants us to "determine" rather than prove, I would be tempted to do it by inspection and argue that since f(x) and x are bounded on the domain of intergation and n is large ...