y=(1+1/n²)^n
两边同时取自然对数得:
lny=nln(1+1/n²)=[ln(1+1/n²)]/(1/n)
lim【n→∞】lny
=lim【n→∞】[ln(1+1/n²)]/(1/n)
=lim【n→∞】(-2/n³)/(-1/n²)
=lim【n→∞】2/n
=0
故lim【n→∞】y=1
即lim【n→∞】(1+1/n²)^n=1
y=(1+1/n²)^n
两边同时取自然对数得:
lny=nln(1+1/n²)=[ln(1+1/n²)]/(1/n)
lim【n→∞】lny
=lim【n→∞】[ln(1+1/n²)]/(1/n)
=lim【n→∞】(-2/n³)/(-1/n²)
=lim【n→∞】2/n
=0
故lim【n→∞】y=1
即lim【n→∞】(1+1/n²)^n=1