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It has no meaning.

If you know L'Hospital's Rule, you can easily construct limits which superficially appear to be indeterminate forms like ±∞/±∞, 0/0, 1^∞, ∞⁰ which can be constructed from actual limits which are +∞, -∞, 0, 1, or any real number.

For example:

lim_x→∞ (1+x)^α/x. Superficially, this looks like 1^∞. But, for every N, no matter how large it may be 1^N = 1. This is pretty much a textbook definition that the idea of 1^∞ = 1. However [you can verify this very quickly with a calculator or by plugging numbers into the google web page search] You can make the above limit any positive real number by choosing α correctly. For example, if α = 1, then lim_x→∞ (1+x)^1/x = e. If α = 0, that limit = 1, If α = -1, it's 1/e. If α = ln(π) lim_x→∞ (1+x)^α/x = π.

You can do the same thing with any indeterminate form that looks like ∞/∞. Just tune the limit properly and it can be 0, ∞, or any other number you like, including the OP's "1." That's why indeterminate forms are symbols, which have no meaning.

The number of reals is infinite. The number of integers is infinite. The statement [number of reals]/[number of integers] has no meaning. The cardinality of the reals is greater than that of the integers. There is no bijective function [1-1 and onto] that maps the integers to the reals. There is a "sense" in which there are more reals than integers, but that "sense" is not translatable in terms of "the number of objects in the sets," because both are infinite.

Infinities are to mathematicians as high current live wires are to electricians: There are tools for handling them, and they won't harm you as long as you're careful. Rigor is the key.

This physicist may want to get rid of the concept of infinity in physics, but he can't. The reason he can't is that he cannot get away from the concept of zero. And as long as you are talking about zero, infinity is always on the other side of that coin.