Is it permissible to euthanize animals if they are old or sick?

If the sick animal is one whose meat is not eaten and there is no hope for its recovery, there is no harm in killing it; because keeping it alive leads to wasting money by spending on it, while it is forbidden to let it die of hunger and thirst. However, if the animal is one whose meat is eaten, and its illness has reached a point where it can no longer be utilized or given to someone who can benefit from it, then it is permissible to kill it by slaughter or shooting, choosing the most merciful method for it.