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 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.
1 answer
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.
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.)
0 comment threads
1 comment thread