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I was wondering if this is true:

$$1.\underbrace{0000\dots0000}_{\infty}1 = 1$$

I tried expressing this as a limit, but I have no idea how to continue:

$$\lim_{x\to-\infty}1+10^{x}$$

Also, I'm not so experienced with limits, etc.

Aditya Hase
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Adnan
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1 Answers1

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As has been pointed out in the comments, this "number" does not exist within $\mathbb{R}$. However, you say that you are stuck with the limits. So I will tell you how to solve the limit:

To solve the limit issue, use the following fact$^{\dagger}$.

$\lim_{x\rightarrow c} f+g=\lim_{x\rightarrow c} f+\lim_{x\rightarrow c} g$.

Here, $\lim_{x\rightarrow\infty}0.1^x=0$, so you go: $$\lim_{x\rightarrow\infty}1+0.1^x=1$$ This is the same as you have, but I rather like going to $+\infty$.

Of course, this just says that $\lim_{x\rightarrow\infty}1+0.1^x=1$, not that $0.\underbrace{00000\cdots00}_{\infty}1=1$

$^\dagger$See, for example, Theorem 2 of section 1.2 (p67) of Adams Calculus: A complete course.

AlanGIC
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    This is not true. $1+\frac1{10}^x$ is not the same as having infinitely many $0$ and then $1$. – Asaf Karagila Dec 01 '14 at 11:40
  • @Asaf True, but I supposed (perhaps incorrectly) that the OP knew this and answered their limit question. Their actual "number" makes no sense, so working within the reals this is really ($\mathbb{R}$eally) the only answer. Unless I am missing something? – AlanGIC Dec 01 '14 at 11:47
  • No, the correct answer is that this is not a real number at all, nor it defines one. (See the discussions in the suggested duplicate.) – Asaf Karagila Dec 01 '14 at 11:48
  • @AsafKaragila Sure, but it depends on how you want to interpret the question. You want to do some serious, high-level maths. I am sitting drinking a cup of coffee and just wanted to point out to someone how to determine a limit (which they say they were stuck with). So I will edit my answer to make this clear. – AlanGIC Dec 01 '14 at 11:50
  • @Integrator sure, but both I and you answered this question a full 10 minutes before the duplicate was pointed out. So how was I to know?!? – AlanGIC Dec 01 '14 at 11:54
  • Alan, asking about the limit is one thing, but since the limit $\lim_{n\to\infty}1+\frac1{10^n}\neq1.\underbrace{0\ldots0}_{\infty\ 0\text{'s}}1$, this is not the right answer to the question. – Asaf Karagila Dec 01 '14 at 11:54
  • @Asaf Yes, I know. I originally would have made this a comment, but lack the rep. However, I do not really understand the issue here, in the sense that the linked answers answer the high-level question, and this answer points out how to solve the OPs (flawed) interpretation using limits. I help with the OPs step 2, and you and others help with the ambient question (step 1). – AlanGIC Dec 01 '14 at 12:02