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Can't Think of a Breaking Bad Pun For the Title: Let's Do Some Math!

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Skewbrow:
Odd degree polynomial over the reals? What can you say about its limits as x goes to +/- infinity?

Loki:
...
Thanks.  :psyduck:

I don't remember what properties of polynomials we did or did not prove, so I might have to prove that your hint is applicable in a fairly roundabout way but that has definitely set me on the right track.

ankhtahr:
Even though I feel bad for bothering you guys with so many exercise sheets of mine, I guess I don't have much of a choice. I really need the points now. (University thread for more info)

So yeah. After missing two lectures because of being sick, I'll have to turn in advanced mathematics tomorrow. The sheet is this one here. We only have to do the tasks with a (K), so 25 and 27.
Sorry Skewbrow, advanced mathematics, not linear algebra. But I have no idea how to even approach these tasks. A friend of mine told me that it would be simple calculating, but he's basically always a week ahead in exercise sheets and I don't know how. My other friends don't really know either. Tomorrow evening (it's evening now as well) we'll be sitting in the university again, trying to do these tasks.

PthariensFlame:
Ok, so I might actually be able to help you here (thank you, Google Translate!).


25 is all about reducing expressions to simplest form; the expressions just happen to contain limits in them, but the limits are incidental to the overall goal.  Most of them require you to invoke l'Hôpital's rule, possibly multiple times:  if f and g are differentiable functions and there exists some k such that f(k) = g(k) = 0 or f(k) = g(k) = infinity, then limx->k f(x)/g(x) = limx->k f'(x)/g'(x).


27 just requires you to inspect the given functions f and g and determine, in each case, what subset of the function's domain it's continuous on.  For example, the absolute value function is continuous over its whole domain, but its derivative is not (it's missing 0).

Loki:
As a semi-helpful hint for 27a, you should know by now that if two functions f(x) and g(x) are continuous in x0, then their combinations are also continuous in x0, ie
1) f(x0)+g(x0)
2) f(x0)-g(x0)
3) f(x0)*g(x0)
4) f(x0)/g(x0) (provided that g(x0) is not 0 of course)
are continuous.

27b: The rational numbers Q are dense in R (please don't beat me up as I wasn't able to find the correct translation for "liegen dicht in R"). Which means that for any number p which is in R\Q, there is a number q in Q which is infinitely close to it (and vice-versa). What values would the function take between, say, 1 and 1 plus some infinitely small ε? Does this look like a continuous function?

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