Sap coming from apricot limb.
Q. Our mature apricot tree has been a good producer of sweet apricots during the 7 years we have owned the house. This year the apricots were not sweet and there is sap coming out of one of the limbs. Now, after harvest, several limbs are dying. I am attaching a photo of the damage. Is there anything we can do to save our tree?
A. Sap coming from the limbs of apricots is usually an indication of stress or damage of some sort. Apricots are not as notorious as peach or nectarine for getting borers or boring insects in their limbs. Damage from borers is typically on the upper surface of limbs because of sunlight intensity or sun damage and it’s very thin bark. Damage from sunburn borers can cause sap to flow near the damaged area.
Your picture seems to indicate sun damage but that can also be from damage from boring insects. By late June or July if this damage were from boring insects or borers you would see a limb which has leaves which have totally turned brown and perhaps limb death. 
Branch death would be a clear indication of borers and of course the limb should be cut out and removed.
There is no chemical spray that you can safely up like to apricots for controlling borers. We usually rely on whitewashing limbs and the trunk with a dilute white latex paint and remove limbs that are heavily damaged. This white latex paint is diluted with an equal amount of water, mixed and applied with a brush or sprayer on the upper surfaces of limbs, western and southern exposed areas of the limbs and trunk. This white wash helps to decrease damage to the limbs from sunburn and subsequent infestation by boring insects.