Sarahisme Posted April 3, 2005 Share Posted April 3, 2005 lol i am having more trouble.... there is always 1 or 2 question i can't do and it drives me nuts!!! well the one this time is.... A function g has the property that lim x→∞ xg(x) = L. Prove that lim x→∞ g(x) = 0. i have never done or been shown how to do a proof like this one and i am compltely lost Sarah Link to comment Share on other sites More sharing options...
Dapthar Posted April 3, 2005 Share Posted April 3, 2005 A function g has the property that lim x→∞ xg(x) = L. Prove that lim x→∞ g(x) = 0. Contradiction seems to be the most straigthforward approach. One can easily prove that [math]\lim_{x\to \infty}{g(x)}[/math] cannot be [math]\infty[/math] or [math]-\infty[/math]' date=' and similarly that it cannot be any finite, non-zero value. Thus, the only case that is left is that if the [math']\lim_{x\to \infty}{g(x)}[/math] does not exist. Thus, assume that [math]\lim_{x\to \infty}{xg(x)}=L[/math], but that [math]\lim_{x\to \infty}{g(x)}[/math] does not exist, and find a contradiction, and then you're done. (You should be able to do this part on your own, just go back to the definition of the existence of a limit.) Link to comment Share on other sites More sharing options...
Sarahisme Posted April 3, 2005 Author Share Posted April 3, 2005 ummm i am a bit confused here..... what you first say.... One can easily prove that \lim_{x\to \infty}{g(x)} cannot be \infty or -\infty, and similarly that it cannot be any finite, non-zero value. Thus, the only case that is left is that if the \lim_{x\to \infty}{g(x)} does not exist. seems to contradict (or something like that) what you next say... Thus, assume that \lim_{x\to \infty}{xg(x)}=L, but that \lim_{x\to \infty}{g(x)} does not exist, and find a contradiction, and then you're done. (You should be able to do this part on your own, just go back to the definition of the existence of a limit.) i am of course wrong but i not sure why yet.... lol its just you first say that it is easy to prove that it doesnt exist, and then you say to prove by contradiction by assuming it doesnt exist and then showing it does....??? maybe its just because its late at night, lol and everything makes no sense at this hour....but anyways... night night Dap Link to comment Share on other sites More sharing options...
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