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Question

If limx0αxexβ loge(1+x)+γx2exxsin2x=10,α,β,γR,
then the value of α+β+γ is

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Solution

Using expansion, we have:
limx0 αx(1+x1!+x22!+)β(xx22+x33+)+γx2(1x1!+x22!+...)x3(sinxx)2=10

limx0x(αβ)+x2(α+β2+γ)+x3(α2β3γ)+...x3=10

αβ=0,α+β2+γ=0,α2β3γ=10

α=6,β=6,γ=9

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