Saturday, 7 December 2013

exponential Limit involving Trigonometric function




limx[cos(2π(xx+1))α]x2

where αQ




Try: l=limx[cos(2π(11x+1))]x2




Put 1x+1=t. Then limit convert into l=limt0(cos(2πt)α)1t1



ln(l)=limt1(1t1)ln(cos(2πt)α)



Could some Help me to solve it, Thanks


Answer



Hint:



Put 1/x=t to find




limt0(cos2πt+1)1/t2



=limt0(cos(2π2πt+1))1/t2 as cos(2πx)=cosx



limt0(cos2πtt+1)1/t2



=[limt0(12sin2πtt+1)12sin2πtt+1]2limt0(2sinπtt+1t)2


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