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

1 answer  ·  posted 3y ago by TextKit‭  ·  last activity 3y ago by The Amplitwist‭

Question calculus
#6: Post edited by user avatar TextKit‭ · 2021-02-19T16:56:20Z (over 3 years ago)
  • 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 by user avatar TextKit‭ · 2021-02-19T16:44:33Z (over 3 years ago)
  • 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 by user avatar The Amplitwist‭ · 2021-02-19T16:43:31Z (over 3 years ago)
added transcript of image, added pointer to the titular equation rather than just "the part in green", minor typographical adjustments
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 by user avatar TextKit‭ · 2021-02-18T16:42:07Z (over 3 years ago)
  • 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 (over 3 years ago)
  • 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 (over 3 years ago)
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