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计算题已知lim趋近于无穷((x+c)/(x-c))^(x/2)=3,求c.
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计算题 已知lim 趋近于无穷 ((x+c)/(x-c))^(x/2)=3,求c.
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答案和解析
lim(x→∞) [(x+c)/(x-c)]^(x/2)
=lim(x→∞) [1+2c/(x-c)]^(x/2)
=lim(x→∞) [1+2c/(x-c)]^{[(x-c)/(2c)]*[(cx)/(x-c)]}
=e^lim(x→∞) [(cx)/(x-c)]
=e^c
=3
∴c=ln3
=lim(x→∞) [1+2c/(x-c)]^(x/2)
=lim(x→∞) [1+2c/(x-c)]^{[(x-c)/(2c)]*[(cx)/(x-c)]}
=e^lim(x→∞) [(cx)/(x-c)]
=e^c
=3
∴c=ln3
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