Continuity mistake: At the end when the helicopter swings in front of the woman on the cliff, the damage to the left side of the helicopter is missing - bullet holes, broken glass etc.
Revealing mistake: When we see Ethan Hunt jump from the helicopter, you can see that it is not Tom Cruise but his stunt double.
Continuity mistake: During the end bike chase scene, Ethan Hunt has just used his mirror to fire his Beretta at Sean Ambrose, he then re-holsters the weapon and swerves round two cars. If you play the movie slowly you can see that after he re-holsters the gun it is still in his left hand while he is holding onto the handlebars. It's a quick shot so you have to look carefully.
Continuity mistake: Near the start of the car chase at the end, a white car loses a hub cap in extreme close-up, but when it wipes out completely and flips over a few seconds later all its hub caps are still on.
Continuity mistake: During the motorcycle scene near the end of the movie there is a shot of Cruise from the front. You can clearly see an earring on his left ear. A second later it's gone.
Answer: Ethan did not want her to kill herself. Giving her a gun would allow her to do so.