Question

lim(x→0)[2^x/㏑(1+2x)-1/㏑(1+2x)]=

Answer 1

lim(x→0) [2^x/ln(1 + 2x) - 1/ln(1 + 2x)]
= lim(x→0) (2^x - 1)/ln(1 + 2x)
= lim(x→0) (2^x - 1)/(2x)、等价无穷小ln(1 + kx) kx
= lim(x→0) [e^(xln2) - 1]/(xln2) * ln(2)/2
= 1 * ln(2)/2
= (1/2)ln(2)
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其中定理lim(x→0) (e^x - 1)/x = 1
证明：设y = e^x - 1,x = ln(y + 1)
lim(x→0) (e^x - 1)/x
= lim(y→0) y/ln(y + 1)
= lim(y→0) 1/[(1/y)ln(y + 1)]
= lim(y→0) 1/[ln(y + 1)^(1/y)]
= 1/ln(e)
= 1