When talking about black holes and singularities, most books say that combining relativity and quantum mechanics gives the answer of infinity in some equations. They also say that:
Infinity is the answer you get when the universe is trying to tell you that you have made a mistake.
Why can't equations have answers of infinity? For example, a singularity is infinitely dense, so why can't other things about it be infinite?
No. Anyone saying that an "infinity is a way of telling you have made a mistake" is being too playful with words. Usually, infinities are a way of telling that you have done something unphysical. But that is not the case with GR -- singularities are a way of telling that you have a spacetime which has some physical uncanniness to it, but one that is physical nonetheless.
When talking about black holes and singularities, most books say that combining relativity and quantum mechanics gives the answer of infinity in some equations.
I am not aware of such books. The reference is usually made to say that quantum gravity might resolve a singularity. I have talked about this at length in many other answers, but the point is that the resolution of a singularity is not physically viable. Especially in a holographic setting, if two independent CFTs are considered, corresponding bulk duals must also be independent. Even if in an ordinary theory, there are no ways to "give the answer of infinities". A singularity is a very meaningful thing with a very precise implication, and singularity theorems are considered to make sense of them physically. If someone states that one can resolve geodesic incompleteness that can be made sense of by a violation of the covariant entropy bound, I would be very surprised, since the holographic description of the entropy flux through a compact (at least) marginally trapped surface would require that the Bousso bound is satisfied. Similarly, with the generalized second law, null generators of the causal domain $\partial \mathcal{J}^{\pm }(I)$ turn out to be incomplete if the GSL is violated, as shown by Wall. At least, in some sense I would expect singularities to not be "resolved". There are claims that loop quantum gravity holds some stance w.r.t resolving singularities, but I for one have not seen a very precise work on this. Do point out references (that are not pop-sci books) for this. See my other answer, https://physics.stackexchange.com/a/781724/365939.
