difference between $+\infty$ and $\infty$

Question

I'm taking Mathematical Analysis "I" and I'm studying limits where I have limits to the infinity, but I don't know what's the difference between $\lim_{x \to \infty}$ and $\lim_{x \to +\infty}$ I guess they are the same but I'm not sure. If you could help me I would appreciate it. Thank you very much!

Answer 1

In the context of real Analysis we usually consider \begin{align*} \lim_{x \to \infty}f(x)\qquad\text{and}\qquad\lim_{x \to +\infty}f(x) \end{align*} to be the same. It has mainly to do with preserving the order of the real numbers when $\mathbb{R}$ is extended by the symbols $+\infty$ and $-\infty$. We look at two references:

• Principles of Mathematical Analysis by W. Rudin.

  Definition 1.23: The extended real number system consists of the real field $\mathbb{R}$ and two symbols $+\infty$ and $-\infty$. We preserve the original order in $\mathbb{R}$, and define \begin{align*} \color{blue}{-\infty < x < +\infty}\tag {1} \end{align*} for every $x\in\mathbb{R}$.

  (he continues with:) It is then clear that $+\infty$ is an upper bound of every subset of the extended real number system, and that every nonempty subset has a least upper bound.

  If, for example, $E$ is a nonempty set of real numbers which is not bounded above in $\mathbb{R}$, then $\sup E=+\infty$ in the extended real number system. Exactly the same remarks apply to lower bounds.

Now we look at certain intervals of real numbers introduced in

• Calculus by M. Spivak.

  (We find in chapter 4:) The set $\{x:x>a\}$ is denoted by $(a,\infty)$, while the set $\{x: x\geq a\}$ is denoted by $[a,\infty)$; the sets $(-\infty,a)$ and $(-\infty,a]$ are defined similarly.

  (Spivak continues later on:) The set $\mathbb{R}$ of all real number is also considered to be an "interval" and is sometimes denoted by \begin{align*} \color{blue}{(-\infty,\infty)}\tag{2} \end{align*}

The connection with limits is presented in chapter 5:

The symbol $\lim_{x\rightarrow\infty}f(x)$ is read "the limit of $f(x)$ as $x$ approaches $\infty$," or "as $x$ becomes infinite", and a limit of the form \begin{align*} \lim_{\color{blue}{x\rightarrow\infty}}f(x) \end{align*} is often called a limit at infinity.

(and later on:) Formally, $\lim_{x\rightarrow\infty}f(x)=l$ means that for every $\varepsilon>0$ there is a number $N$ such that, for all $x$, \begin{align*} \text{if }x>N\text{, then }|f(x)-l|<\varepsilon\text{.} \end{align*}

and we find as exercise 36 a new definition and the following two out of three sub-points

Exercise 36: Define \begin{align*} &\lim_{\color{blue}{x=-\infty}}f(x)=l\\ \\ &(b) \text{Prove that }\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow-\infty}f(-x)\text{.}\\ &(c) \text{Prove that }\lim_{x\rightarrow 0^{-}}f(1/x)=\lim_{x\rightarrow-\infty}f(x)\text{.} \end{align*}

Conclusion: When looking at (1) and (2) together with Spivaks definition of limits we can conclude that $\infty$ and $+\infty$ are used interchangeably in the context of limits of real valued functions.

Answer 2

The compactification of the real numbers, in a useful way that fits in with the ordering of the reals, requires the addition of two points, whereas the compactification of the complex numbers is naturally accomplished by adding just one point. Because analysis readily switches between the real and complex cases, it is considered by some authors appropriate to use a "balanced" pair of symbols, $+\infty$ and $-\infty$, for the real case, which reflects the symmetry of their roles, and the unsigned $\infty$ for the complex case. This is a stylistic choice. Other authors are not of this persuasion. Their argument is "We don't write $+3$ when we mean $3$; so why should we have to write $+\infty$? And, in the complex case, which is always clear from the context, writing $\infty$ is good enough for anyone".

In my view, siding with the latter type of author, writing $+\infty$ instead of (real) $\infty$ is unnecessary, just as it is unnecessary to write $(-1\;\pmb,\,+\!1)$ to denote the interval $(-1\;\pmb,\,1)$.