Tuesday, March 13, 2018

limits - Evaluating limlimitsxtopi/6frac(2sinx+cos(6x))2(6xpi)sin(6x)




limxπ/6(2sinx+cos(6x))2(6xπ)sin(6x)




I would expand with Maclaurin series but xπ6 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:



limy0(2sin(y+π6)+cos(6y+π))26ysin(6y+π)=limy0(3sin(y)+cos(y)cos(6y))26ysin(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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