Monday, 30 October 2017

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



I cannot figure this limit out.




limx0+(ex1)(tanx)23x2



I've used the e to the ln trick and multiplied by 1 (x2x2) and arrived at



limx0+exp(x4/3ln(ex1))



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



I cannot use l'Hospital or Taylor series (only the "known" limits for sin, cos, ex, ln such as limx0sinxx=1 which are really only Taylor series).




Thanks for help!


Answer



Hint: limx0ex1x=limx0tanxx=1


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