Why can we not write: $\lim_{x \to 0}(-\frac{1}{x})=-\infty$

Question

I was skimming through a few books on calculus, and came across the statement that:

we cannot write out this limit as $+\infty$ or $-\infty$ like so: $\lim_{x \to 0}(-\frac{1}{x})=-\infty$, only as $\lim_{x \to 0}(\frac{1}{x})=\infty$. The argument provided was that its a matter of agreement.

Although, it's considered perfectly fine to write this equation likeso: $\lim_{x \to 0}(-\frac{1}{x^2})=-\infty$

My question is whether anyone else has written it in opposition to the statement above, and what is meant by "matter of agreement", and why it may be as such.

Or might it just be the authors own biased opinion, for whatever reason?

I personally see nothing wrong with introducing $+\infty \space or -\infty$ given that it helps to recognise further meaning in supporting your rationale for proving a theorem.

Here's an image of the page where it's taken from:

Reference: Fort, T. (1951) Calculus. pg 50.

Your comments are most welcome.

Answer 1

Presumably, the "matter of agreement" means that denoting $f(x)=-\frac{1}{x}$, you have $$ \lim_{x\to 0+}f(x)=-\infty $$ and $$\lim_{x\to 0-}f(x)=+\infty $$ which do not agree.

Alternatively, the definition of "$\lim_{x\to 0}f(x)=-\infty$" says that

for every $M>0$, there exists $\delta>0$ such that $0<|x-0|<\delta$ implies $f(x)<-M$.

But this is impossible in the case when $f(x)=-\frac{1}{x}$.

Answer 2

If you write, $\ \lim_{x \to 0^+}(-\frac{1}{x})=-\infty,\ $ then I would know what you mean, although I may or may not point out that it is more "proper" to say that the limit " diverges to $\ -\infty $", but some people may say that this is just me being pedantic and I think this is a fair opinion also.

However, saying that $\ \lim_{x \to 0}(-\frac{1}{x})=-\infty,\ $ is just incorrect, because

$\ \lim_{x \to 0^-}(-\frac{1}{x})=\infty \neq -\infty = \lim_{x \to 0^+}(-\frac{1}{x}). $

Answer 3

Without entering the limit definition, ask yourself: how do you approach to $ 0 $?

First

You can take $x=0.00000000...1$, then what do you get?

$$\dfrac 1x=\dfrac {1}{0.00000000...1}=100000000...\longrightarrow \color {red} {\text{positive infinity}}$$

Second

You can take $x=-0.00000000...1$, then what do you get?

$$\dfrac 1x=\dfrac {1}{-0.00000000...1}=-100000000...\longrightarrow \color {blue} {\text{negative infinity}}$$

So, you get two different result:

$$\lim_{x\to 0}\dfrac1x=+ \infty$$

$$\lim_{x\to 0}\dfrac1x=- \infty$$

What will we do now?Which do you think is right? Unfortunately, none of limits are correct. Because, you get two different results. This is called right and left limits in mathematics. These must be equal for the limit to exist. So, we must say "limit doesn't exist."

But, note that this limit is correct:

$$\boxed {\lim_{x\to 0}\dfrac{1}{|x|} =\infty}$$