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I've encountered several limits that need to be solved in order to calculate the coefficients in asymptotic formulas for elementary functions, e.g.:

$$ \lim_{x \to 0}\frac{\sin x -x}{x^3} $$

$$ \lim_{x \to 0}\frac{e^x-1-x}{x^2} $$

Obviously, they're really easy with l'Hopital, but it's illegal since we haven't reached derivatives yet. Are they possible to solve without l'Hopital? Has anybody encountered the same task with asymptotic formulas, i.e.: finding coefficients for higher order asymptotic decompositions while knowing lower order coefficients?

micahyoung
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Van0k
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  • Use the Taylor series expansion for $\sin x$ and $e^x$ – Dylan Nov 28 '15 at 23:13
  • Prohibited as well, since, well, no derivatives) – Van0k Nov 28 '15 at 23:16
  • @tethernova That's a derivative. – zhw. Nov 28 '15 at 23:21
  • @Van0k How else would you find asymptotic formulas for those functions? – Dylan Nov 28 '15 at 23:25
  • Here is a link for the first one: http://math.stackexchange.com/questions/1495624/is-it-possible-to-solve-this-limit-without-hopital-taylor-derivatives The solution provided is very neat – imranfat Nov 28 '15 at 23:26
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    For the second one I am a little bit confused. You have not learned derivatives yet, so how were you introduced to the e-power function? – imranfat Nov 28 '15 at 23:28
  • @Dylan As I stated - recursively. I have to find formulas till x^3 included. I know that sin(x)=a+bx+cx^2+dx^3+o(x^3) and a=0, b=1, c=0, then I'm supposed to find d. – Van0k Nov 28 '15 at 23:30
  • Ok I understand. It wasn't clear at all what you were doing, until now. – Dylan Nov 28 '15 at 23:33
  • @imranfat Much obliged, but I'm afraid I still have just about 15 such limits to solve, and they're all different in terms of used functions. But still thanks a lot) – Van0k Nov 28 '15 at 23:39
  • @Van0k If you need help from the MSE community, it is better to post one problem at a time rather than multiple problems in one post... – imranfat Nov 28 '15 at 23:40
  • @imranfat Thank you for your advice, I've signed up pretty much 5 minutes ago – Van0k Nov 28 '15 at 23:42
  • @imranfat: That solution doesn't show that the limit exists, it only gives the value of the limit if it exist. See here for an alternative: http://math.stackexchange.com/questions/157903/evaluation-of-lim-limits-x-rightarrow0-frac-tanx-xx3/158134#158134. – Hans Lundmark Nov 28 '15 at 23:47
  • @HansLundmark Yes, I understand, but without any derivatives/Taylor this is not easy to do. The OP might not be "there" yet...without the standard tools, it certainly isn't easy for me. – imranfat Nov 28 '15 at 23:50

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I used numerical approximation. For the first of your limits, since $x \rightarrow 0$, I substituted $x=.1,.01,.001,.0001$ etc. The answer seemed to be limiting towards $-0.1666$ repeating, which is $x=-1/6$. I assume the same can be done for the other.

user41728
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