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#2: Post edited by user avatar TextKit‭ · 2021-02-02T21:25:53Z (almost 4 years ago)
  • I'm not asking about [the proof](https://math.stackexchange.com/a/3983012/872474) that I already understand. I'm longing to understand this graphically. As you can see, I added $1/t$ to Stewart's graphs. Then what?
  • [![enter image description here][1]][1]
  • I can ask this as a separate question, but I feel that the underlying difficulty is the same. How can I visualize how Formula 8 shifts to Formula 9 below?
  • [![enter image description here][2]][2]
  • [1]: https://i.stack.imgur.com/6qyPn.jpg
  • [2]: https://i.stack.imgur.com/0cwVM.jpg
  • I'm not asking about [the proof](https://math.stackexchange.com/a/3983012/872474) that I already understand. I'm longing to understand this graphically. As you can see, I added $1/t$ to Stewart's graphs. Then what?
  • [![enter image description here][1]][1]
  • I can ask this as a separate question, but I feel that the underlying difficulty is the same. How can I visualize how Formula 8 shifts to Formula 9 below?
  • [![enter image description here][2]][2]
  • James Stewart, *Calculus* 7th ed 2011. Not Early Transcendentals. p. 443 for the first image.
  • [1]: https://i.stack.imgur.com/6qyPn.jpg
  • [2]: https://i.stack.imgur.com/0cwVM.jpg
#1: Initial revision by user avatar TextKit‭ · 2021-02-02T21:22:23Z (almost 4 years ago)
How can I visualize $\lim\limits_{x \rightarrow \pm \infty} f(x) = \lim\limits_{t \rightarrow 0^{\pm}} f(1/t)$?
I'm not asking about [the proof](https://math.stackexchange.com/a/3983012/872474) that I already understand. I'm longing to understand this graphically.  As you can see, I added $1/t$ to Stewart's graphs. Then what? 

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

I can ask this as a separate question, but I feel that the underlying difficulty is the same. How can I visualize how Formula 8 shifts to Formula 9 below?

[![enter image description here][2]][2]


  [1]: https://i.stack.imgur.com/6qyPn.jpg
  [2]: https://i.stack.imgur.com/0cwVM.jpg