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求lim(x→∞)(3^x +9^x )^(1/X)的极限
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lim
(3^x+9^x)^(1/x)
as
x->∞
=e^lim
ln(3^x+9^x)/x
=e^lim
(3^x*ln3+9^x*ln9)/(3^x+9^x),洛必达法则
=e^lim
(ln3+3^x*2ln3)/(1+3^x),上下除以3^x
=e^ln3*lim
(1+2*3^x)/(1+3^x),提取ln3
=3^lim
(1/3^x+2)/(1/3^x+1),上下除以3^x
=3^(0+2)/(0+1)
=3²
=9
(3^x+9^x)^(1/x)
as
x->∞
=e^lim
ln(3^x+9^x)/x
=e^lim
(3^x*ln3+9^x*ln9)/(3^x+9^x),洛必达法则
=e^lim
(ln3+3^x*2ln3)/(1+3^x),上下除以3^x
=e^ln3*lim
(1+2*3^x)/(1+3^x),提取ln3
=3^lim
(1/3^x+2)/(1/3^x+1),上下除以3^x
=3^(0+2)/(0+1)
=3²
=9
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