Monday, 30 October 2017

calculus - Evaluating limxto0+(ex1)frac(tanx)2sqrt[3]x2



I cannot figure this limit out.




lim



I've used the e to the ln trick and multiplied by 1 (\frac{x^2}{x^2}) and arrived at



\lim_{x \to 0^+} \exp({x^{4/3}} \ln ({e^x-1}))



However I failed at getting further. I tried adding and subtracting \ln x but that got me nowhere.



I cannot use l'Hospital or Taylor series (only the "known" limits for \sin, \cos, e^x, \ln such as \lim_{x \to 0}\frac{sinx}{x}=1 which are really only Taylor series).




Thanks for help!


Answer



Hint: \lim_{x\to 0}\frac{e^x-1}x=\lim_{x\to 0}\frac{\tan x}x=1


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