# Thread: Infinity and the Real Number Line

1. ## Infinity and the Real Number Line

Is infinity an element of the real numbers?

I know that infinity is a concept that denotes larger and larger numbers on the real line, but I was given a definition of a limit as being a finite element of the real numbers but now I have seen examples where the limit of a function has turned out to be infinity. So does this mean that infinity is an element of the reals or is the definition that I was given completely wrong?

2. Originally Posted by Noxide Is infinity an element of the real numbers?

I know that infinity is a concept that denotes larger and larger numbers on the real line, but I was given a definition of a limit as being a finite element of the real numbers but now I have seen examples where the limit of a function has turned out to be infinity. So does this mean that infinity is an element of the reals or is the definition that I was given completely wrong?
What you were given is completely wrong. The concept that $\displaystyle \lim_{x\to c}f(x)=\infty$ is shorthand notation that can wreak havoc on the untrained mind. Saying that $\displaystyle \lim_{x\to c}f(x)=\infty$ does not mean that the limit "equals" some number "$\displaystyle \infty$". What it does mean is that given any $\displaystyle M\in\mathbb{R}$ one can find a $\displaystyle \delta>0$ such that $\displaystyle |x-c|<\delta\implies |f(x)|>M$. In other words, to say that a limit is infinite means you can make $\displaystyle f(x)$ arbitrarily large by making $\displaystyle x$ sufficiently close to $\displaystyle c$.

3. There is an excellent discussion of this confusing topic in Ken Rosss Elementary Analysis textbook.

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