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Question

Evaluate: limx1(ln(1+x)ln2)(3.4x13x)[(7+x)13(1+3x)12]sin(x1)

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Solution

limx1(ln(1+x)ln2)(3.4x13x)[(7+x)13(1+3x)12]sin(x1)

limh0ln(1+1+h2)3[4h+11(1+h][(7+1+h)13(1+3+3h)12.sin(1+h1)

limh0ln(h/2)3[4h(1+h)][(8+h)1/3(4+3h)1/2].sin(h)

limh0ln(1+h/2)2(h/2).3.[4h(1+h)]2{[1+h8×13+.....][1+3h4×12+....]}

limh034[4h1h1]1h[h243h8]

=94ln(4/e).

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