Evaluate: lim(x→1) [ (log(1+x) - log2) * (3 * 4^(x-1) - 3x) ] / [ ((7+x)^(1/3) - (1+3x)^(1/2)) * sin(pi*x) ]

9/pi * (2*log2 - 1)
9/(4*pi) * (log4 - 1)
9/(2*pi) * (log4 - 1/2)
2/(3*pi) * (2*log2 - 1)

Correct answer: 9/(4*pi) * (log4 - 1)

Solution

Each factor is 0 at x=1, giving a 0/0 form. Expanding each to first order in t = x-1 and cancelling t² from numerator and denominator gives the limit.

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