I need help calculating this limit: \(\displaystyle\lim_{{{x}\to\pi}}{\frac{{{\cos{{2}}}{x}+{2}{\left(\pi-{x}\right)}{\sin{{x}}}-{1}}}{{{\left({\cos{{2}}}{x}+{\cos{{x}}}\right)}^{{2}}}}}\) I tried

Samara Richard 2022-04-01 Answered
I need help calculating this limit:
limxπcos2x+2(πx)sinx1(cos2x+cosx)2
I tried to use L'Hospital's rule, but didn't get any result. Can you help me please?
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Answers (2)

Jazlyn Mitchell
Answered 2022-04-02 Author has 14 answers
Let x=t+π
Thus,
limxπcos2x+2(πx)sinx1(cos2x+cosx)2=limt0cos2t+2tsint1(cos2tcost)2=
=limt02tsint2sin2t(1cost)2(2cost+1)2=29limt0tsintsin2t4sin4t2=
=89limt0tsintsin2tt4=89limt0tsintt3=
=89limt01cos{t}3t2=89limt02sin2t23t2=89limt02t243t2=427
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Charlie Haley
Answered 2022-04-03 Author has 14 answers
Hint: breaks it like this
limxπcos2x+2(πx)sinx1(cos2x+cosx)2
=limxπ(cos2x1(πx)2+2sinx(πx))limxπ(πxcos2x+cosx)2
Then use the definition of derivative of a function at x=π
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