Well… they’d go into shock and die. So, there is that.
I’ll say this again. Fighting is not about brute strength; It’s about finesse. This is especially true of sword combat. Loosing an arm is just going to seriously mess with that. If he was using an epee or rapier, and was trained as in some variation of Italian school fencing, he might be able to adjust.
If he lost his dominant arm, I’m inclined to say no. You can learn to compensate for the loss of a limb in day to day life (even if you don’t have access to prosthetic replacements). But, you’ll probably be unable to ever get to a point where you’d actually be effective in combat again. (And I say this while fully aware of Götz von Berlichingen.)
The eye isn’t as serious an issue, especially if it still works. His peripheral vision would be impaired. If he’s effectively blind in one eye, you’re not going to want him trying to shoot things, but it’s not a career ending injury.