Solving Integral Limit: \lim_{n \to \infty } \int_{0}^{1}

[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 !

 

[testito]

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

 

[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?

 

[Hootenanny]

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?

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

 

[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.

 

[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 :

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

can this help you for help me !

i'm really confused !

 

[Hootenanny]

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.

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 ...

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 :

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

can this help you for help me !

i'm really confused !

We will not do your homework for you, but we will help you along the way.

 

[testito]

We will not do your homework for you, but we will help you along the way.

is not a homework however , is just a bizarre exercise that i meet !