This is obviously a fantasy story, so I'm pretty sure we must assume that the character won't end up in a weelchair after the first time he uses his power.

Animals, including humans, anticipate the future results of their muscle activations when moving. When you catch a falling object, you signal to your finger muscles to contract before the optimum moment, because of the delay in signaling and contraction. This isn't something you need to be explicitly trained to do; it just happens as a normal consequence of being in your body all the time. Someone who was born with this superpower would probably take a while to even realize that they had it.

This answer seems to be based on the (odd) premise that limbs are always moving at max speed. If you're constantly moving your normal-speed limbs at max speed all the time, you'll also constantly accidentally hit things. The reason that's not a problem for normal-speed limbs, and also seemingly not for fast limbs, is because you can control how fast your limbs move between stationary and max speed. Fast limbs may make it harder to do precision tasks, but I don't see why there'd be a problem beyond that. Also, it's generally assumed one's physiology can handle using whatever superpower one has.

"3X movement speed but 1X reaction speed is the same as 1X movement speed and 1/3X reaction speed." in which case their opponent has 1/3X movement speed and 1/3X reaction speed, which makes this merely playing around with numbers

@Sneftel "anticipate the future results..." that's an action of the brain, which the OP explicitly said was acting normally. I took the OP at their word. If you think that's not reasonable, please ask the OP for clarification.

@NotThatGuy No, not at all. It's based on the premise that the arm can move faster than the brain is trained to deal with. Although the idea of the arm's transition to greater speed without notice by the brain is an interesting plot twist.

@MS You missed my point. But that's OK.

1) Are you suggesting that you have never made an assumption on this site? Just about every single iteration of superspeed/superstrength has an accompanying increase in ligament/muscle/bone/etc strength, such that characters don't instantly self-destruct. Assuming that to apply here is only natural. 2) The human brain could absolutely adapt to deal with faster limb movement, with most issues being mitigated by training and a small amount of care. Even then, those issues only apply when moving near max speed. 3) It seems most would agree its not a particularly logical point.

@MS (1) No, but I go out of my way to avoid them. The OP did not provide those conditions and unless you can prove to me you can read Ichthys' mind, you don't know the answer. (2) The OP specifically and explicitly said the brain's reaction time and speed were "normal" and continued with the fact that the super, despite the faster abilities of "limbs" (not arms... limbs...) could not walk or run faster than a normal people. I am only taking the OP at his/her word. (3) Incorrect. My point was to help the OP understand that the super is very much open to attack and likely to die. Very logical.

@JBH Sorry, I don’t think I successfully explained myself. Faster-moving muscles don’t require a faster brain to work properly or safely, because you can always just increase the anticipation time. Just like how getting to the airport early doesn’t require fast reflexes; it just requires that you leave earlier.

This is valid if the superpower is introduced recently/abruptly, like in the middle of a fight. If this is a power they've had for some time, however, it can be reliably presumed that their reflexes have since adjusted. The brain's reaction time and its reactions are not the same thing. If there used to be, say, a 300ms gap between moving from a neutral stance to a parry, it would now be 100ms, so the timing of having the arm start/stop moving would change, but that's a matter of training/practice. You'd have to think in terms of smaller motions, admittedly, so precision might be an issue.

@Sneftel That's not how eye-hand coordination works at all. You know, I tried sitting here and moving my hand and "changing the anticipation time" and it doesn't work. The brain must be trained, but that violates a condition set by the OP.

@Palarran I'm just taking the OP at their word. If you allow the brain to train to accomodate the ability, then the OP's condition that the brain has normal speed and reaction times is violated. You can't move faster for shorter periods of time without a faster reaction time.

@JBH Anyone that plays sports regularly can tell you that you're incorrect. Anticipation is a very important skill when it comes to physical activity, even with something as basic as walking down the stairs. Have you ever miscounted the number of steps to the bottom and briefly felt like you were falling? That's due to you not anticipating another step. Accommodating for your increased movement speed through altered anticipation (particularly, effector anticipation) does not require any increase in reaction speed. The two are separate.

@MS I have played sports a great deal in my life. Anticipation isn't magic. The reason batters cannot hit a fast ball isn't because they can't anticipate where the ball will be, but because they can't react fast enough to the pitch. They solve the problem by learning to react faster. The arguments you and others have been promoting demonstrate a lack of comprehension of how the brain works. Anticipation is irrelevant if you can't accommodate the anticipation due to reduced time to react without improving reaction time, which the OP doesn't allow the change.

Your argument is that a person with enhanced speed would constantly be throwing themselves off balance and hitting things because they can't react quickly enough to their own movements. That is, quite frankly, nonsense. You've also chosen quite the ironic example, considering just how important anticipation is to a batter. You might be interested to know that professional batters really don't have much better reaction speed than an average person. Their advantage comes from their ability to anticipate a pitch. Sports scientists have done a great deal of research on the matter.

@MS My argument is that the limits, restrictions, and conditions of the OP trump your analysis, which ignores them.

Are you still pretending anticipation and reaction to be the same thing?

@MS No, and never did I. You can anticipate all you want, but if your brain can't react fast enough to the anticipation then it's irrelevant. Frankly, it's you who's asserting that anticipation and reaction are the same thing. The OP stated very clearly that the brain was normal. That means the brain's controlling signals to the limbs is also normal. Only the limbs are different - which is what prompted my answer. You cannot solve this problem with anticipation alone - and you can't change the brain.

@MS BTW, this argument has gone on long enough. I've participated in editing a book about eye-hand coordination, the limits of foveal vision, and the differences between the parasympathetic nervous system and the sympathetic nervous system. It's obvious you don't understand as much as I do and I'm not even a doctor. I'm going to invite the mods to push this entire discussion into chat. If you feel so passionately that I'm wrong and that my insights have no value to the OP, then you have the privilege of writing your own answer. Please avail yourself of that option.

Simply researching the matter would do you much better than assuming yourself to have rare insight from editing a book discussing tangentially-related topics. I also don't see much value in my answering the question, since I would merely be reaffirming the multiple quality answers provided by people with first-hand experience.