Question

Lim(n→无穷)√n(√(n+1)-√n)

Answer 1

Lim(n→无穷)√n(√(n+1)-√n)
=Lim(n→无穷)√n(√(n+1)-√n)(√(n+1)+√n)/(√(n+1)+√n)
=Lim(n→无穷)√n/(√(n+1)+√n)
=Lim(n→无穷)1/(√(1+1/n)+1)
=1/2

追问

能不能再帮我，因为我课实在听不懂

追答

（4）lim(1/1*2+1/2*3+……+1/n(n-1))=lim(1-1/2+1/2-1/3+……+1/n-1/(n+1))=lim(1-1/(n+1))=1
(6)lim(x→1)(x^2-1)/(2x^2-x-1)=lim(x→1)(x+1)(x-1)/(2x+1)(x-1)=lim(x→1)(x+1)/(2x+1)=2/3
(8)lim(x→6)[√(3x-2)-4]/[√(x-3)-√3]
=lim(x→6)[(√(3x-2)-4)(√(3x-2)+4)(√(x-3)+√3)]/[(√(x-3)+√3)(√(3x-2)+4)(√(x-3)-√3)]
=lim(x→6)[(3x-2-16)(√(x-3)+√3)]/[(x-6)(√(3x-2)+4)]
=lim(x→6)[3(√(x-3)+√3)]/[(√(3x-2)+4)]
=[3(√3+√3)]/[√16+4]
=3√3/4
(10)lim(x→∞ )(2x-1)^30/[(x+2)^15(2x-15)^15]
=lim(x→∞ )(2-1/x)^30/[(1+2/x)^15(2-15/x)^15]
=2^30/2^15
=2^15
(12)lim(x→∞ )(x+cosx)/(3x-sinx)
=lim(x→∞ )(1+cosx/x)/(3-sinx/x)
=1/3
(14)lim(x→∞ )(√(x+p)(x+q)-x)
=lim(x→∞ )(√(x+p)(x+q)-x)(√(x+p)(x+q)+x)/(√(x+p)(x+q)+x)
=lim(x→∞ )((x+p)(x+q)-x^2)/(√(x+p)(x+q)+x)
=lim(x→∞ )((p+q)x+pq)/(√(x+p)(x+q)+x)
=lim(x→∞ )((p+q)+pq/x)/(√(1+p/x)(1+q/x)+1)
=(p+q)/2