Here are three problems I have done on other websites recently, if anyone would care to give them a go, that would be great.

**Problem one**
Compute

$\displaystyle L=\lim_{n\to\infty}\int_{a}^{\infty}\frac{n}{1+n^2 x^2}dx$

**Problem two**
Let $\displaystyle a_1>a_2>a_3\cdots>a_n>0$

Furthermore let $\displaystyle p_1>p_2>p_3\cdots>p_n$

and $\displaystyle p_1+p_2+p_3+\cdots+{p_n}=1$

So then if $\displaystyle F(x)=\left(p_1a_1^x+p_2a_2^x+p_3a_3^x+\cdots+p_na_ n^x\right)^{\frac{1}{x}}$

Compute

$\displaystyle L=\lim_{x\to{0^+}}F(x)$

$\displaystyle L_=\lim_{x\to\infty}F(x)$

$\displaystyle L_2=\lim_{x\to-\infty}F(x)$

**Problem three**
Prove that if $\displaystyle a>1$ that

$\displaystyle 0\leq\int_a^{\infty}\frac{\sin(x)}{x}\leq\int_0^{\ infty}\frac{\sin(x)}{x}~dx$

**Note **
If you have seen my solution, please don't duplicate it, for that wouldn't be fun. Plus, I always want to see other points of view!