In centuries past, the cross of terminal illness was seen as an opportunity to be compassionate to the sick and for "offering up" of suffering for the good of others by the sick. Feelings of depression that can overwhelm us when we become terminally ill, and feelings of grief and sorrow when we see our loved ones afflicted by terminal illness. To be sure, situations of terminal illness can be emotionally difficult, for everyone involved. But feelings are not always an accurate gauge of reality. There is a process of "letting go" and entrustment to God by everyone that needs to happen, a process that must not be short-circuited by euthanasia or physician-assisted suicide.
The truth is that the terminally ill are still children of God, infinitely loved by Jesus our Savior and worthy of every legitimate form of comfort, companionship, care, pain-relief, prayer, and encouragement that we can provide for them. That is the true path of "mercy" at the end of life's journey.
The Church's position is not "keep the person alive for as long as possible, at whatever cost in pain and misery." The Church's position is: Always provide basic human care for every patient. If medical treatment has little chance of significantly improving the patient's condition and/or would be exceptionally painful and burdensome, then such treatment can legitimately be refused—for the person is approaching life's natural end, and therefore, should be given all essential palliative care for the final stage of their life's journey. That is what mercy looks like: It's mercy-dying, not mercy-killing. When we follow Christ's path of mercy, we make room for Him to bring hidden blessings out of the sufferings of terminal illness.