difference between $+infty$ and $infty$

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 suppose that they are the same but I’m not sure. If you could help me I would appreciate it. Thank you very much!

Answer

In the context of real Analysis we usually consider
begin{align*}
lim_{x to infty}f(x)qquadtext{and}qquadlim_{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 $xinmathbb{R}$.

    (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 ${x: xgeq a}$ 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 $lim_{xrightarrowinfty}f(x)$ 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}{xrightarrowinfty}}f(x)
end{align*}
is often called a limit at infinity.

(and later on:) Formally, $lim_{xrightarrowinfty}f(x)=l$ means that for every $varepsilon>0$ there is a number $N$ such that, for all $x$,
begin{align*}
text{if }x>Ntext{, then }|f(x)-l|<varepsilontext{.}
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_{xrightarrowinfty}f(x)=lim_{xrightarrow-infty}f(-x)text{.}\
&(c) text{Prove that }lim_{xrightarrow 0^{-}}f(1/x)=lim_{xrightarrow-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.

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