Infinity is used in a variety of manners. Because it means going on forever, domains and ranges use infinity. For example, the domain and range of the equation y=x are both (-infinity,infinity).
In calculus, infinity is commonly used in limits. This is in one of two ways; either the limit can approach infinity, or the number the limit is of can approach infinity.
Normal models in statistics also use infinity.
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'Infinity' is just a concept of something that is without limit. You find infinity used in math (numbers) and physics (distance or a measurement like time), a lot. Infinity is not a 'place', so cannot be dreadful.
People say that infinity doesn't have a limit because that is the definition of the word infinity. This term is used in the fields of mathematics and of the sciences particularly physics.
Infinity is a term used to denote something with no end. Numbers are infinite as no matter what number you can think of you can always up that number by 1. Thus numbers span into infinity. Infinity itself is not a number.
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1 one infinity divided by infinity