Does undefined always mean infinity?

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Main Question or Discussion Point

While using L' Hospital's rule in evaluating limits, one comes across limits of the following type: $$\lim_{x \to 0} x \ln x$$ Such limits are generally evaluated by taking ##x## to the denominator and make it ##x^{-1}##. In such a case, an indeterminate form ##\frac{\infty}{\infty}## comes, which can be evaluated by L' Hospital's rule.

But logarithmic function is not defined for ##x=0##, because no power can make a number 0.

Books are doing these types of sums by taking ##\ln 0 = \infty##.

But is it correct to say that anything that is undefined, is infinity? Infinity is undefined, but that does not mean that anything that is undefined, is infinity. Infinity is a concept, not a number that we can play with.

N.B.: My problem is not a homework problem. I am asking whether the above concept is actually correct, and it has no direct connection with a particular sum in any book.
 
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  • #3
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  • #4
Svein
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Example: [itex] \frac{d}{dx}\vert x \vert[/itex] is undefined at x=0.
 
  • #5
Stephen Tashi
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Books are doing these types of sums by taking ##\ln 0 = \infty##.
Perhaps you mean that books take ##\lim_{x \rightarrow 0} ln(x) = -\infty##. As you said, the notations "##\infty##" and "##-\infty##" do not denote elements of the set of real numbers. If someone writes "##f(0) = \infty##", the only way to interpret that sensibly is as an abbreviation for ##\lim_{x \rightarrow 0} f(x) = \infty## or a similar statement with a variable name other than "##x##".

An example of something that is undefined in calculus is the notation "##3x + * \sqrt = - / y##". That notation is undefined in the sense that it doesn't denote any particular thing studied in calculus. The use of the word "undefined" in connection with ##\lim_{x \rightarrow a} f(x) = \infty## and similar expressions is more restrictive use of the word. To say ##lim_{x \rightarrow a} f(x)## is undefined means there does not exist a real number ##L## such that ##\lim_{x \rightarrow a} f(x) = L##. This is actually a precise statement because it can be intepreted by negating the definition of ##lim_{x \rightarrow a} f(x) = L##. To say ##Lim_{x \rightarrow a} f(x) = \infty## is even more specific because it says ##f(x)## fails to have a limit at ##a## in a specific way.
 
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  • #6
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Technically, you should never be plugging in the limit point, but evaluating the limit.
I am not plugging in the limit to find an answer, but in order to find the indeterminate form (so that I can apply L' Hospital's rule), I am actually putting in the limiting value mentally.

For example, in the limit $$\lim_{x \to 0} x \ln x$$ if I put in ##0##, I am getting ##0 \cdot \infty##. If I take ##x## to the denominator and make it ##x^{-1}##, I get the form ##\frac{\infty}{\infty}##. I have to assume (at least mentally) that ##\ln 0 = \infty## so that I get the indeterminate form. And that is my question: is it correct to assume ##\ln 0 = \infty## so that I can get a form where I can use L' Hospital's rule?
 
  • #7
PeroK
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I am not plugging in the limit to find an answer, but in order to find the indeterminate form (so that I can apply L' Hospital's rule), I am actually putting in the limiting value mentally.

For example, in the limit $$\lim_{x \to 0} x \ln x$$ if I put in ##0##, I am getting ##0 \cdot \infty##. If I take ##x## to the denominator and make it ##x^{-1}##, I get the form ##\frac{\infty}{\infty}##. I have to assume (at least mentally) that ##\ln 0 = \infty## so that I get the indeterminate form. And that is my question: is it correct to assume ##\ln 0 = \infty## so that I can get a form where I can use L' Hospital's rule?
No. That's not right. You should be evaluating the limits on the numerator and denominator as limits. Note that ##\pm \infty## are valid limits, which correspond to well-defined cases where the limit is not a real number.
 
  • #8
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No. That's not right. You should be evaluating the limits on the numerator and denominator as limits. Note that ##\pm \infty## are valid limits, which correspond to well-defined cases where the limit is not a real number.
Understood. Thanks.
 
  • #9
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Books are doing these types of sums by taking ##\ln 0 = \infty##
For example, in the limit $$\lim_{x \to 0} x \ln x$$ if I put in ##0##, I am getting ##0 \cdot \infty##. If I take ##x## to the denominator and make it ##x^{-1}##, I get the form ##\frac{\infty}{\infty}##. I have to assume (at least mentally) that ##\ln 0 = \infty## so that I get the indeterminate form. And that is my question: is it correct to assume ##\ln 0 = \infty## so that I can get a form where I can use L' Hospital's rule?
You've made the same mistake twice: ##\ln 0## is not ##\infty##. For one thing, 0 is not in the domain of the real-valued natural log function. Also ##\lim_{x \to 0^+}\ln(x) = -\infty##, not ##\infty## as you wrote twice.
 
  • #10
Stephen Tashi
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But is it correct to say that anything that is undefined, is infinity?
No, it is not correct.
(By "anything" did you mean "any limit of a function"? It would also be incorrect to say that anytime the limit of a function is undefined, there must be some mention of infinity. Think about functions that fail to have limits because they oscillate, e.g. ##\lim_{x \rightarrow 0} sin(1/x)##.)

Infinity is undefined
You are correct, but many individual words used in mathematics do not have their own definitions.. For example, the notation "##lim_{x \rightarrow a} f(x) = L## when expressed in words uses the terms "limit" and "approches", but the definition does not define "limit" and "approaches" as individual words. Only the complete sentence "The limit of f(x) as x apporaches a is equal to L" is defined.

but that does not mean that anything that is undefined, is infinity.
Yes, that is correct.

Infinity is a concept, not a number that we can play with.
Yes, infinity is not a number.
Yes, by itself, the word "infinity" indicates a vague concept.

"Infinity" can have a specific mathematical meaning as part of a complete sentence. For example, "##lim_{x \rightarrow a} f(x) = \infty##" has a specific definition.
Likewise ##lim_ {n \rightarrow \infty} f(n) = L## has a specific definition.
 
  • #11
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But is it correct to say that anything that is undefined, is infinity? Infinity is undefined, but that does not mean that anything that is undefined, is infinity. Infinity is a concept, not a number that we can play with.
Although it does not apply well to this example, there is a lot more rigorous mathematics regarding infinity, different magnitudes, and different orders of infinity than you probably would expect.
 
  • #12
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Although it does not apply well to this example, there is a lot more rigorous mathematics regarding infinity, different magnitudes, and different orders of infinity than you probably would expect.
I know that, and I would like to learn some of that type of math in the coming days, so as to clear up my concepts on infinity.
 
  • #13
PeroK
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I know that, and I would like to learn some of that type of math in the coming days, so as to clear up my concepts on infinity.
There are two basic uses of the concept of "infinity" in mathematics:

A) Infinite sets, in the sense of sets that have infinitely many members. In this case, an infinite set is one that is not finite.

The set ##\{1, 2, 3, 4, 5 \}## is finite, as it has ##5## members.

The set ##\mathbb{N} = \{1, 2, 3, \dots \}## is not finite, so it is called an infinite set. Specifically, it is "countably" infinite.

The size of a set in terms of how many members it has is called its Cardinality.

Famously, you can prove that the set of real numbers is "uncountably" infinite, as it has a greater cardinality than the set of natural numbers, and there is a whole sequence of sets of increasing cardinality.

B) Infinity used in the treatment of limits. In this case, infinity is not really a concept in itself, but the symbol ##\infty## is used as part of a mathematical definition.

For example, you have ##\lim_{n \rightarrow \infty} a_n##, denoting the limit of a sequence. In this case, the symbols ##\lim, \rightarrow, \infty## have no independent meaning. It's really only the whole construction that has a meaning, which defines a property of the sequence. Very crudely, it's what happens to the sequence "as n gets large". If we say that:

##\lim_{n \rightarrow \infty} a_n = L##, where ##L## is a real number, then this means precisely (as you probably know):

##\forall \ \epsilon > 0, \ \exists N \ s.t. \ n > N \ \Rightarrow |a_n - L| < \epsilon##

The symbol for infinity is also used as the limit of a sequence:

##\lim_{n \rightarrow \infty} a_n = +\infty##, which means precisely:

##\forall \ M > 0, \ \exists N \ s.t. \ n > N \ \Rightarrow a_n > M##

Again, the symbol ##+\infty## has no independent meaning, but only as part of that precise mathematical construction.
 
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  • #15
stevendaryl
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In lots of cases where there is some function that is undefined at a point, it means that the function "goes to infinity" at that point, meaning that the functions value increases without bound as you approach that point. But not all cases of something being undefined work that way.

For example, ##sin(\frac{1}{x})## near ##x=0##. It's undefined, but it isn't infinite (because ##sin## is never greater than 1).
 
  • #16
mathman
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Any discontinuous function is "undefined" at points of discontinuity, unless specifically defined for that point. "Infinity" is a well defined concept. "Undefined" is used when we don't know for example 0/0.
 

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