n→∞:lim[√(n²+a²)]/n
=lim√[1+(a/n)²]
=lim√(1+0)
=1
令lim0.999.9=x(小数点后n个9) (1)
则:10lim0.999.9=10x
lim9.99.9=10x (2)
n→∞时:(2)-(1)
9x=lim9.99.9-lim0.9999.9
=lim9
=9
x=1
∴lim0.999.9=1
或:n→∞:lim0.999...9=lim[1-0.1^n]=1
n→∞:lim[√(n²+a²)]/n
=lim√[1+(a/n)²]
=lim√(1+0)
=1
令lim0.999.9=x(小数点后n个9) (1)
则:10lim0.999.9=10x
lim9.99.9=10x (2)
n→∞时:(2)-(1)
9x=lim9.99.9-lim0.9999.9
=lim9
=9
x=1
∴lim0.999.9=1
或:n→∞:lim0.999...9=lim[1-0.1^n]=1