Post History
#6: Post edited
- 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.
- >
- > <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
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.
- >
- > <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
- 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
- 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.
- >
- > <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
- 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.
- >
- > <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
- 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
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.
- >
- > <sup>James Stewart, *Calculus Early Transcendentals*, 7 ed. (2011) p. 223.</sup>
- 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
- 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
- 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
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