How can I evaluate \lim_{v \to \frac{\pi}{3}} \frac{1-2\cos v}{\sin(v-\frac{\pi}{3})} without

Osvaldo Apodaca 2021-12-30 Answered
How can I evaluate limvπ312cosvsin(vπ3) without using LHospitals rule?
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Expert Answer

Medicim6
Answered 2021-12-31 Author has 33 answers
You can use the fact that
2cosv=2cos((vπ3)+π3)
=cos(vπ3)3sin(vπ3)
It follows from this that
12cosvsin(vπ3)=1cos(vπ3)sin(vπ3)3
and therefore all that remains to be done is to compute
limt01costsint
and for this you can use the fact that
1cost=2sin2(t2)   and that  sint=2sin(t2)cos(t2)
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temnimam2
Answered 2022-01-01 Author has 36 answers
just use Taylor expansions:
cosx=123(xπ3)2+o(xπ3), xπ3
sinx=xπ3, o(xπ3), xπ3
limxπ312cosxsin(xπ3)=limxπ311+3(xπ3)+o(xπ3)xπ3+o(xπ3)=limxπ33+o(1)=3
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Vasquez
Answered 2022-01-08 Author has 460 answers

Use Prosthaphaeresis fromula and sin2A=2sinAcosA Formulas,
2limvπ3cosπ3cosvsin(vπ3)=2limvπ32sin(v2π6)sin(v2+π6)2sin(v2π6)cos(v2π6)
Now cancel out sin(v2π6) as sin(v2π6)0 as vπ3, vπ3

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