Should one use "tends to" or "equals" when dealing with infinity?

Question

Is this perfectly valid: $$\lim_{x\to0}\frac{1}{x}=\infty \tag{1}$$ or should I use: $$\frac{1}{x}\to\infty\quad\text{as}\quad x\to0 \tag{2}$$ or likewise: $$\lim_{x\to0}\frac{1}{x}\to\infty \tag{3}$$ are all valid? I always got taught you should never put infinity in an equation or as an answer, rather you should say "f(x) tends to infinity" because nothing can "equal" infinity, however my textbook is using the first example, likewise with many resources I've looked at online.

Cheers!

Answer 1

The first notation and the second are both common, the third is never used. Yes, we define $\lim_{x\to x_0}f(x)=\infty$ and we can use the $\infty$ symbol in equations, appropriately. However, the equations themselves are in fact incorrect. The limits $\lim_{x\to 0}\frac{1}{x}$ does not exist, since if we approach $0$ from the left and the right we get different results: $\lim_{x\to 0^+}\frac{1}{x}=\infty$ and $\lim_{x\to 0^-}\frac{1}{x}=-\infty$, where the former means approach $0$ from the right and the latter from the left. The limits are different and hence saying $\lim_{x\to 0}\frac{1}{x}=\infty$ is incorrect.

Answer 2

The first two are valid and mean the same thing - the third makes very little sense. Don't ever write the third, please.

Regarding whether nothing can equal infinity: WE have a definition of what $\lim_{x\to0}1/x=\infty $ means - there are no infinities in that definition.

Answer 3

Good question!

Essentially we have that

$$\frac 1x \to \infty \text { as } x\to 0^+$$

Where $\to$ has an English meaning of "tends to"

And we also have that:

$$\lim_{x\to 0^+}{\frac 1x}=\infty$$

"The limit as x tends to 0 from the positive side of 1 over x is infinity"

In short, use "tends to" when not using the limit notation, use equals when you are.

Answer 4

You were taught wrong. "$\lim_{x \to 0} f(x) = \infty$" has a precise meaning, which is the same as "$f(x) \to \infty$ as $x \to 0$". On the other hand, you should never say $\lim_{x \to 0} f(x) \to \infty$.

Your particular example, though, is wrong. $\lim_{x \to 0} 1/x$ doesn't exist, as it involves both positive and negative values of $x$. $\lim_{x \to 0+} 1/x = +\infty$, but $\lim_{x \to 0-} 1/x = -\infty$.