\lim_{x \to 0} \frac{x(1+a \cos x)-b \sin x}{x^3}=1, how to

hadejada7x 2022-01-16 Answered
limx0x(1+acosx)bsinxx3=1, how to find the constants a,b?
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Vivian Soares
Answered 2022-01-17 Author has 36 answers
If you can use Taylor series, the numerator is
x(1+aax22)b(xx36)+o(x3)=(1+ab)x+(b3a)x36+o(x3)
Solving 1+ab=0,b3a=6 gives a=52,b=32
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kaluitagf
Answered 2022-01-18 Author has 38 answers
Apply LHopitals rule again:
limx0x(1+acosx)bsinxx3=LHRlimx01+(ab)cosxaxsinx3x2
=LHRlimx0(b2a)sinxaxcosx6x
and recall that limx0sinxx=1
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