Sunday, 21 August 2016

calculus - Simple limit with asymptotic approach. Where's the error?



Simply calculus question about a limit.



I don't understand why I'm wrong, I have to calculate
limx02x38x2sinx1cosx3



Using asymptotics, limits and De l'Hospital rule I would write these passages...



=limx0x(238x2)x3/2=limx02338x28x2xx=16



But the answers should be 56. Thank you for your help.



Answer



The mistake lies at the beginning :
limx02x38x2(xx36)1cosx3=56



limx02x38x2x1cosx3=16



At denominator 1cos(x3/2) is equivalent to 12x3. Thus one cannot neglect the x3 terms in the numerator. So, the equivalent of sin(x) must not be x but xx36 . This was the trap of the exercise.


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