Why is heat transfer reversible when temperature difference is infinitesimal? I don't understand why heat transfer from hot reservoir to the system is considered reversible in this case:
$T_{reservoir}$ = $T_{system}$ + dT
but it's considered irreversible in this case:
$T_{reservoir}$ = $T_{system}$ + ΔT
Where dT is infinitesimal difference while ΔT is finite difference in temperature  between reservoir and the system.
In both cases some heat is transferred from the reservoir to the system, so it should be irreversible in both cases. What understanding am I missing here?
 A: It is reversible in the first case because it satisfies the reversibility definition. A thermodynamic process is called reversible if an infinitesimal change of the external condition reverses the process. Consider a system at temperature $T$ in thermal contact with a thermal reservoir at same temperature. By an infinitesimal increase $dT$ of the reservoir's temperature you get heat flowing to the body. With a further infinitesimal decrease, let us say $2dT$ you reverse the flow. The same will not happen if there was a finite difference of temperature.
A: To do it reversibly, you can heat the body from $T_1$ to $T_2$ (i.e., over a finite temperature change) using an infinite sequence of constant temperature reservoirs, in which each reservoir in turn is only dT higher in temperature than the body at any time (and also only dT higher in temperature than the reservoir before it in the sequence).  Each increment in heat transfer would take place with only a differential temperature driving force between the body and the current reservoir. To reverse the process, and bring both the body and the reservoirs back to their original states, you would simply contact the body with the reservoirs in the reverse sequence, in which case the reservoirs would be dT lower in temperature than the body in each step of the process).  The only difference would be with regard to the very first and very last reservoirs (which could not be returned to their original states).  But this would be insignificant.
In the case where the body is heated from $T_1$ to $T_2$ by bringing it into contact with a constant temperature reservoir at $T_2$ for the entire time until the body equilibrated at $T_2$, all the heat transfer would take place with a finite temperature driving force, and there would be no way to return both the body and the reservoir to their original states without causing a significant change in something else (like using other reservoirs).
A reversible process is one in which the system is only slightly removed from being at thermodynamic equilibrium throughout the change.  Thus, a reversible process can be viewed as imposing a continuous sequence of thermodynamic equilibrium states.
A: Because the effects of heat transfer.
If the difference in temperature is large, the process can't be reversed. Heat is a form of energy that can transform into an increase of the molecules' movement in a system and if the system is water, and this water is hydrostatic, the kinetic energy will become pressure. So to reverse this, you need to do a refrigeration process which requires work and so it affects the surroundings, so the process then can't be reversed.
If heat transfer is due to infinitely small amount of temperature difference, the process can be reversed without refrigeration process and keeping the surrounding intact.
Now transferring an amount heat through an infinitesimal temperature difference would require an infinite amount of time, so in other words, rate of $Q$ transferred is infinitely small, making it possible for us to reverse the process because the temperature of the lower temperature body and higher temperature body would stay constant. So no kinetic energy and the process here is isothermal!!
So here you go 
Now about about the example you shown, one process is reversible and the other is not even though they both goes through the same process and pass exactly the same state. But one is reversible internally and externally, the other would affect the surroundings.
A: Short answer is that there are irreversibilities associated with both the heat transfers. But the irreversibility associated with heat transfer by a finite temperature difference $ΔT$ is so much larger than the irreversibility associated with heat transfer by an infinitesimal temperature difference $dT$, that the latter process can be approximated as reversible. 
Simply said the extent of irreversibility associated with the $dT$ case is so small that it can be approximated as reversible.
A: Because we would like to have a quasistatic process as it is a condition for reversible process.. So if temperature difference is more then process will not be quasistatic, but if temperature difference is Infitesimali small then the heat transfer will take place in very large time making our process quasistatic (almost static) Fullfilling the condition of Reversibiliyty.....
Now why quasistatic process are reversible that's a different topic in itself..
Thank you
