((1^x+3^x+5^x)/3)^(1/x)趋于0的极限

Question

((1^x+3^x+5^x)/3)^(1/x)趋于0的极限

Answer 1

lim[x→0] [(1^x+3^x+5^x)/3]^(1/x)
= lim[x→0] e^ln[(1^x+3^x+5^x)/3]^(1/x)
= e^lim[x→0] ln[(1^x+3^x+5^x)/3]^(1/x)
= e^lim[x→0] ln[(1^x+3^x+5^x)/3] / x
=洛必达法则= e^lim[x→0] [(1^x+3^x+5^x)/3]'/[(1^x+3^x+5^x)/3] / 1
= e^lim[x→0] [(ln1*1^x+ln3*3^x+ln5*5^x)/3] / [(1^x+3^x+5^x)/3]
= e^{ [(ln3*3^0+ln5*5^0)/3] / [(1^0+3^0+5^0)/3] }
= e^{ [(ln3+ln5)/3] / 1 }
= e^[ ln15^(1/3) ]
= 15^(1/3)