Limit in Definition of Riemann Integral is one-sided?
Let me answer that question with another example, and I hope it clears things up.
We write $$\lim_{x\to 0} \sqrt{x}=0$$ although $$\lim_{x\to 0^{+}} \sqrt{x}=0$$
would also be okay. The first you will see more often.
If you want me to be more explicit:
When discussing limits, we only care about the domain of the function. In this case $\Delta x_k>0$. So it really IS a two sided limit. It's just that there is no "left side" to consider.