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Did Stewart prove the result in green, himself or as an exercise, in Calculus Early Transcendentals or the normal version Calculus?

I dislike duplicates as much as the next guy. A result like this, from first year calculus, must have gotten asked here. But I can't find similar posts.

I attempted this, 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 $xm$.

By the way, if I was the one writing the solution, I would've started with the game plan: turning $(1 + x/n)$ into the form $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{n \rightarrow \infty} (1 + 1/n)^n$. And I wouldn't have changed variable to $m$. Just rewrite $(1 + x/n)$ as $(1 + \dfrac{1}{\frac nx}).$ But if you craved change of variable, I would let $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$.

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

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James Stewart, Calculus Early Transcendentals 7th ed 2011. p. 231.

  • Since $x$ is positive and isn't function of $n$, yes : $m = n/x $. It's basic operation on limits. – purple Jan 12 '21 at 23:37
  • I think you may not be parsing the (somewhat ambiguous and misleading statement) correctly. What is meant is that "if $m = n/x$ then $m$ (as a function of $n$) tends to $\infty$ as $n$ tends to $\infty$; also $n = mx$, therefore:". – Rob Arthan Jan 12 '21 at 23:40
  • @RobArthan isn't my question's title the same thing as what you wrote? $m= n/x \iff n = mx$. –  Jan 12 '21 at 23:46
  • Don't rely on the title of your question to state the question: include a complete statement of the question in the body. In this case, the answer to the question in the title is "yes, provided $x > 0$". I was trying to answer the question in the body of your question which is about the statement you highlighted in green. – Rob Arthan Jan 12 '21 at 23:51

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