Simply calculus question about a limit.
I don't understand why I'm wrong, I have to calculate
limx→02x−3√8−x2sinx1−cos√x3
Using asymptotics, limits and De l'Hospital rule I would write these passages...
=limx→0x(2−3√8−x2)x3/2=limx→0233√8−x28−x2xx=16
But the answers should be 56. Thank you for your help.
Answer
The mistake lies at the beginning :
limx→02x−3√8−x2(x−x36)1−cos√x3=56
limx→02x−3√8−x2x1−cos√x3=16
At denominator 1−cos(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 x−x36 . This was the trap of the exercise.
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