Abusing Notations: Is it unacceptable to write $\lim\limits_{t \to \infty} f(t) = \infty$?

Question

Suppose $f:[0, \infty) \to \mathbb{R}$, $f(t)$ is a function that diverges as $t \to \infty$ i.e. $f(t) = e^{t}$

It is very common to see people (mainly beginning math students) write

$\lim\limits_{t \to \infty} f(t) = \infty$

I don't think this is good practice, since we are equating something to something else that is not a number. What would be a better way to write this?

I would personally write:

$f(t) \to \infty$, as $t \to \infty$

Is it okay to also write

$\lim\limits_{t \to \infty} f(t) \to \infty$

Answer 1

It is very common to see people (mainly beginning math students) write

$\lim\limits_{t \to \infty} f(t) = \infty$

This is fine, and can be made completely rigorous. See extended real number line.

Of course it could still be used in a wrong way but to write $\lim_{t \to \infty} t^6 - 67 = \infty$ is perfectly correct.

I would personally write:

$f(t) \to \infty$, as $t \to \infty$

That's also fine. I would consider it as "synonym" of the above.

Is it okay to also write

$\lim\limits_{t \to \infty} f(t) \to \infty$

No, this is a strange mix of notations. The limit, when it exists, is some fixed 'thing' (a real number or a 'symbol' $\infty$ or maybe $-\infty$ or still something else). In any case, the limit does not approach anything it is (equal to) something (or does not exist).

Answer 2

Just an addition to @quid's answer, it is actually not unusual to see such notations and definitions in many analysis books (and school manuals) including wikipedia. These definitions are rigorous and should not be confused with divergence though, e.g. $f(x)=x \cdot \sin{x}$ and $x_n=\pi \cdot n$, $y_n=\frac{\pi}{2} + 2n\cdot \pi$ and $z_n=\frac{\pi}{2} + (2n+1)\cdot \pi$.

Answer 3

Another addition to @quid's answer: as said, $\lim_{t\to\infty} f(t)=\infty$ can be properly defined with the extended real number line. But you can also make another choice, of not writing what does not exist: for instance, you don't write $\frac{3}{0}$ because this number does not exist, so you can choose not to write $\lim_{x\to\infty}x\sin(x)$ because $x\sin(x)$ has no limit in $\infty$, or $\int_{-\infty}^{\infty} x^2\ \mathrm{d}x$, same for $\sum_{i=1}^\infty \sqrt{i}$. So even if $\lim_{t\to\infty} f(t)=\infty$ is not incorrect, I would personally prefer something else, such as the alternatives proposed.

Answer 4

The point I want to emphasize here is: $\lim f(t)=\infty$ is a notation we use to stand for some specific behavior of a function; it is not declaring the equality of two numbers.

The math community has decided over time to use the notation $\lim f(t)=\infty$. In a parallel universe, parallel-we might have decided to use some different notation; but for us, this is the standard notation, and is therefore correct by definition. I believe we have chosen to use this notation because the "ambiguity" with the standard meaning of the equals sign, on balance, helps our understanding more than it misleads us. But that is a judgment call, not a mathematical fact.

While one can talk about extended real number lines, nonstandard analysis, etc., I don't think they're relevant to the real discussion here. In standard calculus, we use the notation $\lim f(t)=\infty$, but that doesn't mean that infinity is a number. That's simply not what the notation means (according to our standard conventions). Indeed, if I see that notation, I would not say that the limit exists—I would say that the limit fails to exist, in a very specific way (for example, unlike the ways in rtybase's answer). When a function "diverges to infinity", that means the limit doesn't exist, but also that the manner of divergence is a specific one that we recognize as important enough to have given it its own notation.

(I've specifically avoided the subscripts on the limits above, because the discussion applies just as well to $\displaystyle\lim_{t\to\infty}$ and $\displaystyle\lim_{t\to3}$ and so on.)