Question

一条求函数的极限的题目求以下函数的极限:lim[(x^2+x)^1/2-(x^2-x)^1/2]=?(x→∞）

Answer 1

(x^2+x)^(1/2)-(x^2-x)^(1/2)

=[(x²+x)-(x²-x)]/[(x^2+x)^(1/2)+(x^2-x)^(1/2)]

=2x/[(x^2+x)^(1/2)+(x^2-x)^(1/2)]

=2/[(1+1/x)^(1/2)+(1-1/x)^(1/2)]

lim{(x^2+x)^(1/2)+(x^2-x)^(1/2)}

=lim 2/[(1+1/x)^(1/2)+(1-1/x)^(1/2)]

=2/(1+1)

=1