Question

第五题和第七题怎么做？？求过程。。

Answer 1

(5)
lim [√(5x-4)-√x]/(x-1)
x→1
=lim [√(5x-4)-√x][√(5x-4)+√x]/(x-1)[√(5x-4)+√x]
x→1
=lim [(5x-4)-x]/(x-1)[√(5x-4)+√x]
x→1
=lim 4(x-1)/(x-1)[√(5x-4)+√x]
x→1
=lim 4/[√(5x-4)+√x]
x→1
=4/[√(5·1-4)+√1]
=4/(1+1)
=2
(7)
lim [√(x²+x)-√(x²-x)]
x→+∞
=lim [√(x²+x)-√(x²-x)][√(x²+x)+√(x²-x)]/[√(x²+x)+√(x²-x)]
x→+∞
=lim [(x²+x)-(x²-x)]/[√(x²+x)+√(x²-x)]
x→+∞
=lim 2x/[√(x²+x)+√(x²-x)]
x→+∞
=lim 2/[√(1+ 1/x)+√(1- 1/x)]
x→+∞
=2/[√(1+0)+√(1+0)]
=2/(1+1)
=1