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The definition of limit says that Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ If....{the rest of definition is left to make the question easier}.

What does the phrase "except possibly at $a$ itself" mean? What is the significance of defining the interval, open i.e why not closed?

Charlie Parker
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M Shehzad
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  • For example, x/x is not defined at x=0. ​ ​ –  Jul 30 '16 at 15:46
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    It makes no difference if the interval is closed, as long as it has $a$ in its interior. – André Nicolas Jul 30 '16 at 15:47
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    @AndréNicolas : $\ldots,$which is the same as saying there's some suitable open interval. $\qquad$ – Michael Hardy Jul 30 '16 at 15:51
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    To make the open/closed statement even clearer - a closed (or half open) interval would allow the possibility that it ends at $a$, eg takes a form such as: $[a, a+1]$ or $(a-1, a]$. These only contain values on one side of $a$. But the standard definition of a limit requires that the value is the same when a is approached from any direction. – David E Jul 30 '16 at 18:56
  • I don't understand the need for vague titles when your question content is extremely precise and clear. Suggested title: What does the phrase "except possibly at $a$ itself" mean in the definition of a limit? – Charlie Parker Jul 06 '18 at 16:33

3 Answers3

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Here's an answer at more of a calculus level and not so much a real analysis level.

Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself.

"except possibly at $a$ itself" means the function may not actually be defined at $x = a$. And a function need not even be defined at $x=a$ in order for the limit as $x \to a$ to exist. This is because the limit of $f(x)$ as $x \to a$ describes how $f(x)$ behaves near $x=a$ and not necessarily at $x = a$.

For example, $f(x) = \frac{x^2-4}{x-2}$ is not defined for $x=2$. But we can calculate the limit as follows: $$\lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x\to 2} (x+2) = 4$$

This means that as $x \to 2$, our function $f(x)$ looks (behaves) like it's approaching the value $f(x) = 4$. And again, the function isn't even defined at $x = 2$, but this is irrelevant for the limit.

To really drive home the point that the value of the function at $x=a$ is irrelevant to the limit as $x \to a$, consider the following three functions:

  • $f(x) = \frac{x^2-4}{x-2}$
  • $g(x) = x^2$
  • $h(x) = \begin{cases} x+2, & x \ne 2 \\ 10, & x = 2\end{cases}$

All three of these functions have the same limit as $x \to 2$. The limit is $4$. But note the following:

  • $f(x)$ is not even defined at $x=2$.
  • $g(x)$ is defined at $x=2$ and $g(2) = 4$, which is the same value as the limit.
  • $h(x)$ is defined at $x=2$ but $h(2) = 10$, which is not the same value as the limit.
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    +1 for the abstraction level matching the OPs. You could add $x \sin(1/x)$ near $x=0$ as a less artificial example for the case illustrated in your first bullet point. – Ethan Bolker Jul 30 '16 at 18:18
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More generally the definition of limit of a function $f$ at a point $a$ requires that $a$ is a limit point for the domain of $f$; see for example Rudin's book third edition definition 4.1.

When $a$ is a limit point for the domain of a function $f$, you are sure that it's always possible evaluate $f(x)$ when $x\rightarrow a$; this make well defined the expression:

\begin{equation} \lim_{x\rightarrow a} f(x) \end{equation}

Now remember that if $a$ is a limit point of a set $D$, it may be that $a\in D$ or not.

Many books choose to define limit for functions without the concept of a limit point of a set; so they write conditions to bypass the gap. When your book says "except possibly at $a$ itself", it simply reflects the property of a limit point of a set to belong or not to the set itself.

For the second answer: your book choose the point $a$ in an open interval in order to make possible $x\rightarrow a$ from left or right.

If you choose $a$ to be in a closed interval, it may be possible that $a$ is an end point; for example $a\in [a, b]$; but in this case you can make $x\rightarrow a$ only from right and not from left; this make the definition of "right limit" and not the definiton of limit in general.

Hope this help.

rgnnt
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  • what about the case when a ∈ (a,b) , how can we make x close to a from the left? there are no other elements to be on the left of a? If you can explain this to me I'd really appreciate it.Thank you! – Maths Survivor Dec 26 '17 at 12:54
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An open interval that contains $a$ also contains all points sufficiently close to $a$. That is why an open interval is used in the definition.

"except possibly at $a$ itself" was included because the limit of the function at $a$ is determined by the behavior of the function near $a$ but not at $a$.

Michael Hardy
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    No. The fact that the value at $a$ is irrelevant follows from the part of the definition he left out, specifically $0<|x-a|<\delta$. The phrase "except possibly at $a$ itself" means that $f(x)$ need not even be defined for $x=a$. – David C. Ullrich Jul 30 '16 at 16:36
  • @DavidC.Ullrich : And the reason $a$ was omitted from the domain was that that gets done in the part that he left out. $\qquad$ – Michael Hardy Jul 30 '16 at 19:48
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    Huh? No, that does not get done in the part he left out. And more to the point, that has nothing to do with the question. He asked what the "except possibly at $a$" meant. What you said is not what that means. – David C. Ullrich Jul 30 '16 at 19:50
  • @DavidC.Ullrich : I explicitly gets done in the part that was omitted, but the fact that that was to be done is the reason why $a$ was excluded from the domain. $\qquad$ – Michael Hardy Jul 30 '16 at 19:57
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    The question was not why $a$ gets ommitted from the domain! (It's not even true that $a$ gets omitted from the domain, but let's not worry about that.) The question was what "except possibly for $a$ itself" means. If you'd said something about why $a$ need not be in the domain you might have said something true, that just didn't answer the question. But you said that "except possibly at $a$ itself" means that the limit is determined by the behavior near $a$ but not at $a$ itself. What you said is simply wrong. False. Not true. The meaning of that phrase is not what you said it is. – David C. Ullrich Jul 30 '16 at 20:09
  • @DavidC.Ullrich : ok, I've reprhased it. And $a$ does get omitted from the part of the domain that's relevant to the definition. $\qquad$ – Michael Hardy Jul 30 '16 at 20:23