已知x→∞时lim(2x-c)/(2x+c)∧x=3,求c

Question

已知x→∞时lim(2x-c)/(2x+c)∧x=3,求c

Answer 1

x→∞
lim [(2x-c)/(2x+c)]^x
=lim [(2x+c-2c)/(2x+c)]^x
=lim [1+(-2c)/(2x+c)]^x
=lim e^ln([1+(-2c)/(2x+c)]^x)
=lim e^ x*ln[1+(-2c)/(2x+c)]
=e^ lim x*ln[1+(-2c)/(2x+c)]

因此,
ln3
=lim x*ln[1+(-2c)/(2x+c)]
利用等价无穷小：ln(1+x)~x
=lim (-2cx)/(2x+c)
=lim (-2cx)/x / (2x+c)/x
=lim -2c / (2+(c/x))
=-2c / 2
=-c
因此,解得：c=-ln3
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