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Comments on If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$?

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If $n = xm$ and $n \rightarrow \infty$, then $m \rightarrow \infty$?

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Did Stewart prove the titular result, also underlined in green in the image below, either by himself or as an exercise, in Calculus Early Transcendentals or in the normal version Calculus?

I attempted the proof, but I got nonplussed. $\infty = \lim\limits_{n \rightarrow \infty} n = \lim\limits_{n \rightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $x$ and $m$.

  1. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.

James Stewart, Calculus Early Transcendentals  7 ed 2011. p. 223.

Transcription of image follows below:
56. Let $m = n/x$. Then $n = xm$, and as $n \to \infty$, $m \to \infty$.

Therefore, $\displaystyle \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = \lim_{m \to \infty} \left( 1 + \frac{1}{m} \right)^{mx} = \left[ \lim_{m \to \infty} \left( 1 + \frac{1}{m} \right)^m \right]^x = e^x$ by Equation 6.

James Stewart, Calculus Early Transcendentals, 7 ed. (2011) p. 223.

By the way, if I was writing the solution, I would've started with the game plan: rewriting $(1 + x/n)$ in the form $(1 + 1/\mathrm{something})^{\mathrm{something}}$ in order to apply $e = \lim\limits_{something \rightarrow \infty} (1 + 1/\mathrm{something})^{\mathrm{something}}$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx}\right).$

But if you pined to change variable, I would've commenced with defining $\dfrac xn$ as $\dfrac 1 m$ which is more intuitive than "let $m = n/x$", because only $\dfrac xn$ shows up explicitly in the question. The question doesn't manifest $n/x$.

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General comments (5 comments)
General comments
Peter Taylor‭ wrote over 3 years ago · edited over 3 years ago

Do you have a definition of $\lim_{a \to \infty} f(a) = \infty$ in first order logic (i.e. as a simple statement with $\exists$ and $\forall$)?

TextKit‭ wrote over 3 years ago
r~~‭ wrote over 3 years ago

That won't do; that definition requires the limit to be a finite number, but you're working with an infinite limit.

Also, downvoted because once again, you're using images of text instead of writing your question such that no images are required.

TextKit‭ wrote over 3 years ago

@r~~ Do you mean https://i.imgur.com/i5jxtFW.jpg? "you're using images of text instead of writing your question such that no images are required" Images ARE required. I don't want to mis type the solution manual. I want to show you exactly what it says.

The Amplitwist‭ wrote over 3 years ago

Just transcribe the content of the image in addition to posting the image. It's good that you want to ensure that you're not mistyping the solution manual, but presenting text as images makes it difficult or impossible for people with accessibility issues to know what you're referring to.