From a Biblical point of view, it is a faulty premise to say that Adam and Eve were exempt from death. Most Christians believe that the soul is immortal because this is what they have been taught. However the Bible sustains that a soul can die. God says:
The soul that sinneth, it shall die. The son shall not bear the
iniquity of the father, neither shall the father bear the iniquity of
the son: the righteousness of the righteous shall be upon him, and the
wickedness of the wicked shall be upon him.
In fact, it was the devil's first lie.
And the serpent said unto the woman, Ye shall not surely die.
On the other hand is the quotation from God.
But of the tree of the knowledge of good and evil, thou shalt not eat
of it: for in the day that thou eatest thereof thou shalt surely die.
There are many profitable reasons why this belief came about. However the Bible sustains that a person that is dead simply sleeps until the Second Coming.
For the living know that they will die; But the dead know nothing, And they have no more reward, For the memory of them is forgotten.
Jesus describes death as sleep.
These things said he: and after that he saith unto them, Our friend
Lazarus sleepeth; but I go, that I may awake him out of sleep.
Then said his disciples, Lord, if he sleep, he shall do well.
Howbeit Jesus spake of his death: but they thought that he had spoken
of taking of rest in sleep.
Then said Jesus unto them plainly, Lazarus is dead.
Paul describes death like sleep and clearly shows that people who die, wait until resurrection at the Second Coming.
But I do not want you to be ignorant, brethren, concerning those who
have fallen asleep, lest you sorrow as others who have no hope. For if
we believe that Jesus died and rose again, even so God will bring with
Him those who sleep in Jesus.
For this we say to you by the word of the Lord, that we who are alive
and remain until the coming of the Lord will by no means precede those
who are asleep. For the Lord Himself will descend from heaven with a
shout, with the voice of an archangel, and with the trumpet of God.
And the dead in Christ will rise first. Then we who are alive and
remain shall be caught up together with them in the clouds to meet the
Lord in the air. And thus we shall always be with the Lord. Therefore
comfort one another with these words.
1 Thessalonians 4:13-18
Biblically, Adam and Eve COULD die. So Biblically, if you believe they could not die, God would have been lying to them and the devil would be telling the truth(See Genesis 3:4 and Genesis 2:17 above). Jesus Himself tells us that the devil is the father of lies.
You are of your father the devil, and the desires of your father you
want to do. He was a murderer from the beginning, and does not stand
in the truth, because there is no truth in him. When he speaks a lie,
he speaks from his own resources, for he is a liar and the father of
Adam and Eve COULD die like their Creator told them and this is why they were taken out of Eden.
Then the Lord God said, “Behold, the man has become like one of Us, to
know good and evil. And now, lest he put out his hand and take also of
the tree of life, and eat, and live forever”
In order to live forever, Adam and Eve must eat from the tree of life, according to the Scripture above. This is what the tree of life is for.