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 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: When Sean removes his Ethan mask on the 747 his eyes change from Ethan's clear blue to his own dark. But there is no eye cover or contact lenses incorporated in the mask. (00:04:40)
Continuity mistake: At the rock climbing sequence, after he slips and is hanging by one hand starts staring at the landscape, in the wide shots, the sun is directly behind him, but in the closeup shots, he is in the shadow. (00:07:45)
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.