Random process $X(t)$ with autocorrelation function given find the mean and the variance
The limit $\lim_{\tau\to\infty} R_x(\tau)$, if it exists, equals $E^2[X(t)]$ and so $E[X(t)]=0$ in this case. 

More generally, the mean of a WSS process is nonzero only if the power spectral density has an impulse at the origin. This can be applied to 
periodic autocorrelation functions such as $\cos(\omega_0t)$ pointed out
in @MattL's comment. If the Fourier series for a periodic autocorrelation 
function has a nonzero DC term, the mean is nonzero.