Does limit 1/x at zero equal infinity? How it is accepted in High School now?

Hello. Per what I was taught in my youth, ##\lim_{x \to 0}\frac{1}{x}=\infty##

Is it in agreement with how the calculus is taught today in the High Schools and Universities of US/Canada specifically?
Per what my son says, that limit should be considered as undefined because

##\lim_{x \to 0^+}\frac{1}{x}=+\infty##
and
##\lim_{x \to 0^-}\frac{1}{x}=-\infty##

and as these infinities have different signs, the general limit does not exist (even if expressed as "infinity").

Disclaimer: I understand that my question is about the conventions and about how the math is taught, not about the math itself.

Are you saying that in your youth you were taught that ##\lim_{x \to 0}(\pm\frac{1}{x})=\infty##?
In my youth I was taught that ##\lim_{x \to 0}(\pm\frac{1}{x})=\pm \infty.##

Furthermore, if I saw ##\lim_{x \to 0}(\frac{1}{x})##, I would assume an implicit ##+## sign in front of the fraction. The fact that we are taking a limit does not invalidate that common assumption. I think your son is nitpicking.

No. I said that I was taught that ##\lim_{x \to 0}\frac{1}{x}=\infty##
I.e. no plus/minus sign before ##\frac{1}{x}##

And in the other two lines

##\lim_{x \to 0^+}\frac{1}{x}=+\infty##
and
##\lim_{x \to 0^-}\frac{1}{x}=-\infty##

I meant one-sided limits, a right-hand and a left-hand limits, i.e.

##\lim_{x \to (0^+)}\frac{1}{x}=+\infty##
and
##\lim_{x \to (0^-)}\frac{1}{x}=-\infty##

> I think your son is nitpicking.

I simply asked to understand how the mainstream High School and University textbooks treat this limit. I.e. what should be the right answer.

MichPod said:

##\lim_{x \to (0^+)}\frac{1}{x}=+\infty##

##\lim_{x \to (0^-)}\frac{1}{x}=-\infty##

The first line in your quoted statement says that if ##x## is on the positive x-axis and approaches zero right to left, the fraction ##\frac{1}{x}## goes to plus infinity.

The second line in your quoted statement says that if ##x## is on the negative x-axis and approaches zero left to right, the fraction ##\frac{1}{x}## goes to minus infinity.

Now what about the bone of contention, ##\lim_{x \to 0}\frac{1}{x}=\infty##? If we go by the convention that ##x## without a sign means some distance from the origin on the positive axis, then the statement is correct. If we understand that ##x## is just a placeholder and can have positive or negative values, then your son is correct and the statement is ambiguous.

However, limiting expressions such as this are not standalone and devoid of context. When we take a limit, we know on which side of the axis we are and how we approach the origin. Thus, we know whether the first or second line in your statement above is the case.

MichPod said:

> I think your son is nitpicking.

I simply asked to understand how the mainstream High School and University textbooks treat this limit. I.e. what should be the right answer.

I don't think that what you (or your son) say is wrong. It's just dotting all the i's and crossing all the t's when everybody knows what is meant.

You are using the projectively extended real line and your son is using the extended real number line.

You may like to consider one-sided limits explaining why the central limit doesn't exist using the latter set but it does using the former.

The latter set is much more relevant today because:

https://en.wikipedia.org/wiki/Signed_zero said:

The IEEE 754 standard for floating-point arithmetic (presently used by most computers and programming languages that support floating-point numbers) requires both +0 and −0. Real arithmetic with signed zeros can be considered a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞; division is only undefined for ±0/±0 and ±∞/±∞.

> You are using the projectively extended real line

Well, that could be treated that way, but practically in the approach I was taught, having 1/x limit equal to infinity just meant that the absolute value of ##\frac{1}{x}## may be made arbitrary large in a small enough neighborhood of x=0. That is, in that approach it was not the case that there was only one "infinity", but the following three equations had a defined (and different) meaning:
##\lim_{x \to 0}f(x)=\infty##
##\lim_{x \to 0}f(x)=-\infty##
##\lim_{x \to 0}f(x)=+\infty##

Nevertheless, I am not looking to see who is right here, me or my son or which approach is more relevant. I simply wanted to understand what is ##\lim_{x \to 0}\frac{1}{x}## as per US/Canada mainstream textbooks.

MichPod said:

I simply wanted to understand what is ##\lim_{x \to 0}\frac{1}{x}## as per US/Canada mainstream textbooks.

According to Wolfram, it doesn't exist as stated here:

https://en.wikipedia.org/wiki/Limit_of_a_function#Existence_and_one-sided_limits said:

Alternatively, ##x## may approach ##p## from above (right) or below (left), in which case the limits may be written as

$$\lim _{x\to p^{+}}f(x)=L$$

or

$$\lim _{x\to p^{-}}f(x)=L$$

respectively. If these limits exist at ##p## and are equal there, then this can be referred to as the limit of ##f(x)## at ##p##. If the one-sided limits exist at ##p##, but are unequal, then there is no limit at ##p## (i.e., the limit at ##p## does not exist). If either one-sided limit does not exist at ##p##, then the limit at ##p## also does not exist.

Your son is correct and that is how it should have always been taught. The limits from above and below 0 are very different.

MichPod said:

Hello. Per what I was taught in my youth, ##\lim_{x \to 0}\frac{1}{x}=\infty##

If x approaches 0 from below, the values become large negative numbers. From above they become very large positive numbers. So that should not be called a limit.

MichPod said:

Is it in agreement with how the calculus is taught today in the High Schools and Universities of US/Canada specifically?
Per what my son says, that limit should be considered as undefined because

##\lim_{x \to 0^+}\frac{1}{x}=+\infty##
and
##\lim_{x \to 0^-}\frac{1}{x}=-\infty##

and as these infinities have different signs, the general limit does not exist (even if expressed as "infinity").

Correct.

MichPod said:

Disclaimer: I understand that my question is about the conventions and about how the math is taught, not about the math itself.

It's not just a matter of convention. For values of x very near 0, the negative and positive values of 1/x are not close to each other. So saying that it has a limit would be misleading as to the behavior of 1/x near x=0.

jack action said:

According to Wolfram, it doesn't exist as stated here:

Wolfram is a very good reference, thank you!

Your quote from Wikipedia is probably not so good as it speaks about a limit equal a particular value, while limits which are equal to infinity are special cases and normally need even a separate definition of what they are. Yet at the same page it mentioned that 1/x has no limit on the extended real line, so, again, that shows to me that 1/x having no limit is a very acceptable in practice point of view. Thank you again!

If you plot ##1/x## in the neighborhood of 0, it should be clear that the limit ##x \rightarrow 0## is undefined.

FactChecker said:

Your son is correct and that is how it should have always been taught. The limits from above and below 0 are very different.

Amen.
##\lim_{x \to 0}\frac 1 x## does not exist; i.e., the limit is undefined.

DrClaude said:

If you plot ##1/x## in the neighborhood of 0, it should be clear that the limit ##x \rightarrow 0## is undefined.

Exactly.

kuruman said:

In my youth I was taught that ##\lim_{x \to 0}(\pm\frac{1}{x})=\pm \infty.##

The usual notation is ##\lim_{x \to 0^+}## and ##\lim_{x \to 0^-}## to indicate whether the limit is from the right or left, respectively.

kuruman said:

Furthermore, if I saw ##\lim_{x \to 0}(\frac{1}{x})##, I would assume an implicit ##+## sign in front of the fraction. The fact that we are taking a limit does not invalidate that common assumption.

This doesn't make sense. The expressions ##\frac 1 x## and ##+\frac 1 x## (or ##\frac {+1}x##) are exactly the same.

kuruman said:

I think your son is nitpicking.

Not at all. The son is correct. Perhaps the OP is misremembering what he learned years ago.

Mark44 said:

Perhaps the OP is misremembering what he learned years ago.

Or perhaps I am misremembering what I didn't learn many years ago.

As @FactChecker noted, the question isn't a matter of convention but of definitions. Here's a site with precise definitions for various types of limits.

https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx

##1/x## as ##x \to 0## satisfies neither definition 4 nor 5, so the limit doesn't exist.

Because the question has been answered, I'm closing this thread.