The first sort of objection one would make to orthodox Christology would be like this:
God cannot X
Christ actually did/ was X
Therefore, Christ is not God.
The X might be any number of predicates, viz. die, be born, have a body, learn things, be a creature, not know things, etc.
The orthodox response to the objection is that it fails to grasp what we mean by a hypostatic union, sc. the unity of a human nature in a divine person. The basic principle is this: when something is unified to a person, what affects it affects the person, even if the person can exist without it. If you touch my hand, you touch me; if you cut off my hand, you do me injury even though my person can survive without a hand and might even exist before it (say, in utero.) This principle applies not only to body parts but even to property, clothing or likenesses. If you venerate a statue you venerate the person it is a likeness of, and if you can express your anger or disgust at a person by harming what they own.
There are important ways in which the hypostatic union has to be divided from all these modes of unity to a person, since the hypostatic union is sui generis, but the basic fact of unity-to-person remains. There are also, of course, important objections that can be raised to this response, but it still counts as a response to the first sort of objection given above.