求极限limx→∞[(x+1)2/3-(x-1)2/3]．
2024-08-23高等数学（工专）(00022)
求极限limx→∞[(x+1)2/3-(x-1)2/3]．
【正确答案】：limx→∞[(x+1)2/3-(x-1)2/3] =limxx→∞1/3[(x+1)1/3+(x-1)1/3]•[(x+1)1/3-(x-1)1/3] =limxx→∞1/3[(x+1)+(x-1)}1/3]． [(x+1)-(x-1)]/[(x+1)2/3+(x+1)1/3•(x-1)1/3+(x-1)2/3] =2limx→∞x1/3•[(x+1)1/3+(x-1)1/3]/[(x+1)2/3+(x+1)1/3•(x-1)1/3+(x-1)2/3 =2limx→∞[(1+1/x)1/3+(1-1/x)1/3]/[(1+1/x)2/3+(1+1/x)1/3•(1-1/x)1/3+(1-1/x)1/3] =4/3
【正确答案】：limx→∞[(x+1)2/3-(x-1)2/3] =limxx→∞1/3[(x+1)1/3+(x-1)1/3]•[(x+1)1/3-(x-1)1/3] =limxx→∞1/3[(x+1)+(x-1)}1/3]． [(x+1)-(x-1)]/[(x+1)2/3+(x+1)1/3•(x-1)1/3+(x-1)2/3] =2limx→∞x1/3•[(x+1)1/3+(x-1)1/3]/[(x+1)2/3+(x+1)1/3•(x-1)1/3+(x-1)2/3 =2limx→∞[(1+1/x)1/3+(1-1/x)1/3]/[(1+1/x)2/3+(1+1/x)1/3•(1-1/x)1/3+(1-1/x)1/3] =4/3
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