How often is $x\to\infty$ used to denote ($x\to +\infty$ or $x\to -\infty$)?

Question

How often is $x\to\infty$ used to denote ($x\to +\infty$ or $x\to -\infty$)?

Both my textbook and my teacher use $x\to\infty$ as above, so e.g. it's false for us that $\lim_{x\to\infty}\frac{1}{\sqrt{x}}=0$ and the limit does not exist here.

But here on M.SE everyone means $x\to +\infty$ when writing $x\to\infty$ and this is how Wikipedia defines it, so I think our notation is unusual. Is it? Do mathematicians ever use it?

Answer 1

In some teaching traditions (for example in Italy), there are three kinds of infinity. I'm sure this tradition is not very popular nowadays, but still there are teachers who follow it, in high schools if not at universities.

I was taught limits in high school distinguishing between $+\infty$, $-\infty$ and “unsigned $\infty$):

• $\lim\limits_{x\to+\infty}f(x)=l$ means that, for every $\varepsilon>0$, there exists $M>0$ such that, for all $x>M$, $|f(x)-l|<\varepsilon$;

• $\lim\limits_{x\to-\infty}f(x)=l$ means that, for every $\varepsilon>0$, there exists $M>0$ such that, for all $x<-M$, $|f(x)-l|<\varepsilon$;

• $\lim\limits_{x\to\infty}f(x)=l$ means that, for every $\varepsilon>0$, there exists $M>0$ such that, for all $x$ with $|x|>M$, $|f(x)-l|<\varepsilon$.

• $\lim\limits_{x\to a}f(x)=+\infty$ means that, for every $N>0$, there exists $\delta>0$ such that, for all $x$ with $0<|x-a|<\delta$, $f(x)>N$;

• $\lim\limits_{x\to a}f(x)=-\infty$ means that, for every $N>0$, there exists $\delta>0$ such that, for all $x$ with $0<|x-a|<\delta$, $f(x)<-N$;

• $\lim\limits_{x\to a}f(x)=\infty$ means that, for every $N>0$, there exists $\delta>0$ such that, for all $x$ with $0<|x-a|<\delta$, $|f(x)|>N$.

Here $a$ is supposed to be in $\mathbb{R}$. I'll leave out the similar definition of infinite limit at one of the three kinds of infinity. Usually, very little care was used in stating what the domain of the function was supposed to be.

So we had to learn nine different definitions, just for allowing to say that $$ \lim_{x\to0}\frac{1}{x}=\infty $$ (unsigned infinity) or that $\lim_{x\to\infty}x^3=\infty$.

Of course, the fact that, setting $$ f(x)=\begin{cases} 1/x & \text{for $x\ne0$ rational}\\ -1/x & \text{for $x$ irrational} \end{cases} $$ one can say, according to the definitions above, that $$ \lim_{x\to0}f(x)=\infty $$ didn't ring any bell in the brains of those teachers.

Of course there is some method in this madness: for rational functions, the notion of “unsigned infinity” makes sense, because it just denotes the point at infinity of a line parallel to one of the axes. However, what this is useful for, other than for confusing the students' ideas, I've yet to discover.

As soon as we abandon rational functions, such silly distinctions serve absolutely no purpose. In the case under examination, the limit at unsigned infinity (according to the definition above) might make sense or not, depending on how the requirements about the domains of the functions are established.

Under some conventions, when one says “for all $x$ such that $[\dots]x[\dots]$, we have $[\dots]f(x)[\dots]$” the condition “$x$ in the domain of the function” is subsumed. Under other conventions, the function must be defined in some (punctured) neighborhood of the point under examination.

There's no general consensus, as far as I know. It mostly depends on the level the material is presented to.

In any case, the statement that $\lim_{x\to\infty}1/\sqrt{x}$ doesn't exist might be true (according to some established convention), but doesn't seem to convey useful information about the function.

Answer 2

Yes, $x\to \infty$ usually means $x\to +\infty$. What you mean is usually denoted by $|x|\to \infty$. But of course it isn't wrong if your teacher uses other notation as long as its well defined at the start of the course.

Answer 3

In fact, $x \to + \infty$ and $x \to -\infty$ are different things. For example, given $L \in \Bbb R$:

• $\lim_{x \to +\infty}f(x) = L$ means that for all $\epsilon > 0$ exists $x_0 \in \Bbb R$ such that $x > x_0$ implies $|f(x)-L|<\epsilon$.

• And $\lim_{x \to -\infty}f(x) = L$ means that for all $\epsilon > 0$ exists $x_0 \in \Bbb R$ such that $x < x_0$ implies $|f(x)-L| < \epsilon$.

The only situation I can think now where $x \to +\infty$ would cause no ambiguity, is if $x$ would denote a complex variable - in $\Bbb C$ we don't have $+\infty$ nor $-\infty$, just $\infty$, saying that $\lim_{z \to +\infty} f(z) = L$ to mean that $\lim_{z \to 0}f(1/z) = L$.

Answer 4

It's not unusual to write $x\to\infty$ when $x\to+\infty$ is meant, provided the context makes the meaning clear. There is a substantial difference between $\pm\infty$ in things like $$ \lim_{x\to+\infty} \frac 1 {1+2^x} = 0\ \text{ and }\lim_{x\to-\infty}\frac 1 {1+2^x} = 1. $$ In expressions like $\lim\limits_{x\to\pi/2} \tan x = \infty$, it makes sense to regard this $\infty$ as the $\infty$ that is at both ends of the line, making the line (topologically) a circle, or to regard $\infty$ as the $\infty$ that is approached by allowing ones distance from $0$ in the complex plane to approach $+\infty$ regardless of what direction one is going, thus making the plane (topologically) a sphere. Likewise in $\lim\limits_{z\to\infty}\dfrac{5z+3}{z-2}=5$, the $\infty$ is neither $+\infty$ nor $-\infty$.

However, it would be a mistake to think that a distinction between $+\infty$ and $-\infty$ has no place in complex analysis, since in expressions like $$ \sum_{k=M}^N\cdots\cdots, $$ one can let $M\to-\infty$ or $N\to+\infty$.