Limit of Integral Evaluation: Tips and Tricks

[xeno_gear]

How would I go about evaluating something like this?

$$\[ \lim_{n\to+\infty} n \int_0^{+\infty} \dfrac{\sin\left(\dfrac{x}{n}\right)}{x(1+x^2)}\, dx \]$$

 

[arildno]

Let I(n) be the definite integral.

Rewrite your expression as :
$$\frac{I(n)}{\frac{1}{n}}$$

Note that you can use L'hopitals rule here, and that you may interchange the operation of differentiation with respect to "n" and integration with respect to "x".

 

[arildno]

Another way might be to write:
$$\sin(\frac{x}{n})=\frac{x}{n}++++$$

Inserting, and simplifying, the limit will be the same as the above.

 

[Gib Z]

$$\frac{ n \sin\left(\frac{x}{n}\right)}{x(1+x^2)} \rightarrow \frac{1}{1+x^2}$$ uniformly, so we can bring everything inside the integral.

 

[xeno_gear]

Thanks everyone!