# How can I visualize $\lim\limits_{x \rightarrow \pm \infty} f(x) = \lim\limits_{t \rightarrow 0^{\pm}} f(1/t)$?

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I'm not asking about the proof 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?

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?

James Stewart, *Calculus* 7th ed 2011. Not Early Transcendentals. p. 443 for the first image.

## 6 comments

You should endeavor to put as much of your question as possible in the form of text/MathJax. This makes the question more accessible, e.g. to those using screen readers or who have custom fonts/text size such as for dyslexia or because they have difficulty reading small text. It also makes the question easier for search engines to index. As a bonus reason, I use a dark mode extension, but your question asks me to strain my eyes staring at blazing white boxes to figure out what you're asking. — Derek Elkins about 1 month ago

More specifically to your question, I have no idea what you are trying to communicate with the first image. As far as I can tell, you've simply added the text "$1/t, t \neq 0$ to it. Also, doesn't simply plotting $x = 1/t$ as a function of $t$ not already make it graphically and intuitively obvious that $t$ approaching $0^\pm$ causes $x$ to approach $\pm\infty$? Other properties that might be useful for the purposes of limits such as continuity and monotonicity are also evident from the graph. — Derek Elkins about 1 month ago

@DerekElkins The first image pictures the LHS, i.e. $\lim\limits_{x \rightarrow \pm \infty} f(x)$. How can I visualize the RHS, or why the LHS = RHS, from this picture alone? — Technologically illiterate about 1 month ago

@DerekElkins "doesn't simply plotting $x = 1/t$ as a function of $t$ not already make it graphically and intuitively obvious that $t$ approaching $0^\pm$ causes $x$ to approach $\pm\infty$" Perhaps...but how can I visualize this change of variable right in this image, without plotting another graph? — Technologically illiterate about 1 month ago

@DerekElkins "your question asks me to strain my eyes staring at blazing white boxes to figure out what you're asking." Apology! But I can't render pictures into text or MathJax??? — Technologically illiterate about 1 month ago

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