Integrals I: Feynman's Trick · Superchallenge problem

For every positive real $x$, let $f(x) = \displaystyle\lim_{r\to0}\left(\frac{(x+1)^r + x^r}{2}\right)^{1/r}$. Find $\displaystyle\lim_{x\to\infty}\big(f(x) - x\big)$.