# Monotone conitnuous function - find limits

1. Mar 17, 2008

### daniel_i_l

1. The problem statement, all variables and given/known data
if f is monotonic and continues in R, and $$\int^{\infty}_{a} f(x)dx$$ converges then
$$lim_{x \rightarrow \infty} xf(x) = 0$$

2. Relevant equations

3. The attempt at a solution
I know that if xf(x) converges at all then it has to convergs to 0. But how do I know that it converges?
Thanks.

2. Mar 17, 2008

### Dick

Can you show that for any positive value of k however small, that f(x)<k/x for all x>N for N sufficiently large? Hint: suppose it's not.

3. Mar 18, 2008

Thanks.