(however admittedly I'm awful at math- I'm presuming the second answer is the same?? I'm likely wrong however and advertising my mathematical blondeness! :))
I think the answer is also infinity to the second one (though whether it's positive or negative infinity depends on whether x approaches 5 from below or from above).
"lim" refers to the limit (http://mathworld.wolfram.com/Limit.html) of a function (warning: maths-heavy explanation!).
I think essentially it's used when you can't evaluate a function for a given x, but if you take numbers that are closer and closer to that x, and the function tends towards a given value the closer you get to x, then that value is the limit of f(x) for that x.
The words make sense and the pretty pictures make sense, but I wouldn't have the foggiest idea how to put it into use or why you would use it in the real world :).
Hence my problem with algebra in general. It was never that I couldn't do it, as long as I closed my eyes, blindly followed the rules and hope for the best. I just couldn't rely on the 'rules' I had to know why everything did it, so I could work it in my head, and be able to work it out from different directions to check. Doesn't matter how many Math People I went to nobody could tell me the 'why'.
Actually, you can use a limit even when you can evaluate a function for a given x; it's just that most of the time, you wouldn't want to bother. Like, the limit of f(x)=x^2 as x approaches 2 is 4, but big fat hairy deal, right?
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.
Ah! I assumed you had image loading off in your browser rather than LiveJournal image placeholders (in which case, you're not shown the ALT text, of course).
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
![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)
no subject
(however admittedly I'm awful at math- I'm presuming the second answer is the same?? I'm likely wrong however and advertising my mathematical blondeness! :))
\lim{x \to 5} \frac{1}{x-5}
Re: \lim{x \to 5} \frac{1}{x-5}
\lim
I think essentially it's used when you can't evaluate a function for a given x, but if you take numbers that are closer and closer to that x, and the function tends towards a given value the closer you get to x, then that value is the limit of f(x) for that x.
Re: \lim
The words make sense and the pretty pictures make sense, but I wouldn't have the foggiest idea how to put it into use or why you would use it in the real world :).
Hence my problem with algebra in general. It was never that I couldn't do it, as long as I closed my eyes, blindly followed the rules and hope for the best. I just couldn't rely on the 'rules' I had to know why everything did it, so I could work it in my head, and be able to work it out from different directions to check. Doesn't matter how many Math People I went to nobody could tell me the 'why'.
But nevertheless, I love that example!
Re: \lim
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.
no subject
image ALT text
I tried to make the ALT text a usable "alternative" to the image, so I'm wondering whether anyone ever uses it.
Re: image ALT text
Re: image ALT text
no subject
no subject
no subject
no subject