# difference between \$+infty\$ and \$infty\$ Code Answer

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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!

In the context of real Analysis we usually consider
begin{align*}
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.