![After explaining to a student, with various lessons and examples, that \[ \lim_{x \to 8} \frac{1}{x - 8} = \infty \] I tried to check if she really understood that, so I gave her a different example. This was the result: \[ \lim_{x \to 5} \frac{1}{x - 5} = ['5' rotated 90° to the left] \]](https://p2.dreamwidth.org/b2f97b50a1e6/20791-344478/files.rsdn.ru/187/limit.jpg)
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Philip Newton
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Re: \lim
Date: Wednesday, 20 October 2004 08:42 (UTC)And whenever anything in a limit involves infinity, the definition of "limit" has to change subtly. (Normally, it involves the result "being closer within any particular given tiny number" to the limit; but for the infinite case, it involves "being larger than any particular given huge number".)
And yes, the answer isn't really infinity in the above case, because it varies based on whether x > 8 (or 5 or whatever) or x < 8. Technically, there is no limit of 1/(x-8) as x tends to 8, because of that. (The Weierstrass definition of the limit that you referred to above doesn't address "which direction x comes from"; that was part of the problem with developing a proper definition of limit.)
Sorry if the above is babbly and verbose, but that's what you get when an almost-math-major* who also enjoys the history of mathematics reads your post. %-)
* I have a B.Sc. with a major in computer science, but if I'd hung around for one more semester and taken Calculus of Complex Variables I'd have obtained a double major in math and cs. But I couldn't be bothered. My university didn't have "minors" as such, but it's safe to say I minored in math.