limx→π/6(2sinx+cos(6x))2(6x−π)sin(6x)
I would expand with Maclaurin series but x→π6 so I cannot do that. So I evaluated it with l'Hopital rule (result is −112), but is there a better way? I had to differentiate two times and it gets really big and complicated.
Answer
You can shift the argument with x=y+π6:
limy→0(2sin(y+π6)+cos(6y+π))26ysin(6y+π)=limy→0(√3sin(y)+cos(y)−cos(6y))2−6ysin(6y).
Now it is enough to see that Taylor expansion will give (√3y+o(y))2=3y2+o(y2) in the numerator and −36y2+o(y2) in the denominator, and the limit is
−336.
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