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Q&A

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?

enter image description here

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

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

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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 5 months 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 5 months 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? VLDR‭ 5 months 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? VLDR‭ 5 months 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??? VLDR‭ 5 months ago

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1 answer

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I'm not sure if it's really possible to visualize it without creating another graph, but what you could do is take a function, like f(x) = sin(2x²)/2x², and graph f(x) and f(1/x) on the same graph. Notice how the peaks and valleys of the red function, f(x), and the blue function, f(1/x), correspond to each other.

Graphs of f(x) = sin(2x^2)/2x^2 and also f(1/x), made with Desmos

You could do that double plot with g(x) = (1+x)^{1/x}, although the graph would be more boring and it would be harder to see what is going on.

(If the g(x) expression says "Math input error", then it's supposed to read (1+x)1/x.)

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