The Nazi ideology saw the Jewish blood as a contagion. They believed that allthough extermination of the Jews might be difficult, it was ultimately a good because it would eliminate this contagion. The following is an excerpt of SS leader Heinrich Himmler's famous Posen Speech of October 4, 1943:

I ask of you that that which I say to you in this circle be really only heard and not ever discussed. We were faced with the question: what about the women and children? – I decided to find a clear solution to this problem too. I did not consider myself justified to exterminate the men – in other words, to kill them or have them killed and allow the avengers of our sons and grandsons in the form of their children to grow up. The difficult decision had to be made to have this people disappear from the earth. For the organization which had to execute this task, it was the most difficult which we had ever had.

Throughout the process of exterminating the Jews, infants were killed via shootings, gassings, starvation, bayoneting, etc.

Source

Lewis, Jon. Voices from the Holocaust. London: Robinson, 2012.

I just finished Auschwitz by Dr. Miklos Nyiszli and he describes multiple times the way the prisoners were sorted. Right meant survival (if you want to call it that) and left meant death. If you were fit for work then you went right. Unfit for work or too young to work mean you went left. Nyiszli said boys and girls up to 15 were told to go left. If a mother had a child they both went left.

He did mention one circumstance where a mother (who went right) begged to go with her 14 year old daughter (who went left) and the request was granted. The doctor considered this an act of kindness.

Nyiszli was a coroner for Dr. Mengel. Mengel instructed Nyiszli to write the cause of death for all subjects as a natural one or from various diseases that claimed the lives of those who lived in the camp.