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#6: Post edited by $user avatar$ TextKit‭ · 2021-02-19T16:56:20Z (about 3 years ago)

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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 $m$.
>56. 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.][1]][1]
>
> *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 $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x ightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx} ight).$
~~But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".~~
[1]: https://i.stack.imgur.com/WYuLX.jpg

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 $m$.
>56. 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.][1]][1]
>
> *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 $e = \lim\limits_{something ightarrow \infty} (1 + 1/\mathrm{something})^{\mathrm{something}}$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx} ight).$
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 $n/x$.
[1]: https://i.stack.imgur.com/WYuLX.jpg

#5: Post edited by $user avatar$ TextKit‭ · 2021-02-19T16:44:33Z (about 3 years ago)

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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 ightarrow \infty} n = \lim\limits_{n ightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $xm$.~~
>56. 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.][1]][1]
>
> *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 $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x \rightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx}\right).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

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 ightarrow \infty} n = \lim\limits_{n ightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $x$ and $m$.
>56. 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.][1]][1]
>
> *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 $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x \rightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx}\right).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

#4: Post edited by $user avatar$ The Amplitwist‭ · 2021-02-19T16:43:31Z (about 3 years ago)
added transcript of image, added pointer to the titular equation rather than just "the part in green", minor typographical adjustments

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

~~Did Stewart prove the result in green, himself or as an exercise, in *Calculus Early Transcendentals* or the normal version *Calculus*?~~
~~I attempted the proof, but I got nonplussed. $\infty = \lim\limits_{n ightarrow \infty} n = \lim\limits_{n ightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $xm$.~~
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
>[![enter image description here][1]][1]
~~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 $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x ightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $(1 + \dfrac{1}{\frac nx}).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

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 ightarrow \infty} n = \lim\limits_{n ightarrow \infty} xm $. Now what? The limit is for $n$, but my variables are $xm$.
>56. 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.][1]][1]
>
> *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 $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x ightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $\left(1 + \dfrac{1}{\frac nx}\right).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

#3: Post edited by $user avatar$ TextKit‭ · 2021-02-18T16:42:07Z (about 3 years ago)

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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 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 $xm$.
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
>[![enter image description here][1]][1]
~~James Stewart, *Calculus Early Transcendentals* 7th ed 2011. p. 231.~~
By the way, if I was writing the solution, I would've started with the game plan: rewriting $(1 + x/n)$ in the form $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x \rightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $(1 + \dfrac{1}{\frac nx}).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

Did Stewart prove the result in green, himself or as an exercise, in *Calculus Early Transcendentals* or 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 $xm$.
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
>[![enter image description here][1]][1]
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 $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x \rightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $(1 + \dfrac{1}{\frac nx}).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

#2: Post edited by $user avatar$ TextKit‭ · 2021-02-18T06:42:42Z (about 3 years ago)

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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 $m = n/x$.
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
>[![enter image description here][1]][1]
James Stewart, *Calculus Early Transcendentals* 7th ed 2011. p. 231.
[1]: https://i.stack.imgur.com/WYuLX.jpg

Did Stewart prove the result in green, himself or as an exercise, in *Calculus Early Transcendentals* or 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 $xm$.
>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$.
>[![enter image description here][1]][1]
James Stewart, *Calculus Early Transcendentals* 7th ed 2011. p. 231.
By the way, if I was writing the solution, I would've started with the game plan: rewriting $(1 + x/n)$ in the form $(1 + 1/n)^n$ in order to apply $e = \lim\limits_{x \rightarrow \infty} (1 + 1/x)^x$. It's superfluous to change variable to $m$. Just rewrite $(1 + x/n)$ as $(1 + \dfrac{1}{\frac nx}).$
But if you craved change of variable, I would have commenced with defining $\dfrac xn$ as $\dfrac 1 m$, which is more intuitive than "let $m = n/x$".
[1]: https://i.stack.imgur.com/WYuLX.jpg

#1: Initial revision by $user avatar$ TextKit‭ · 2021-02-18T06:39:06Z (about 3 years ago)

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

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 $m = n/x$. 


>56. Show that $\lim\limits_{n \rightarrow \infty} (1 + \dfrac xn)^n = e^x$ for any $x > 0$. 

>[![enter image description here][1]][1]


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


  [1]: https://i.stack.imgur.com/WYuLX.jpg

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