Question

大一高数求极限的问题，求好人解答，最好有过程

Answer 1

追问

谢谢

Answer 2

(1)
lim(n->∞)[√(n+1) -√n]/[√(n+2) -√n]

consider
y=1/x

lim(x->∞)[√(x+1)-√x]/[√(x+2) -√x]
=lim(y->0)[√(1/y+1) -√(1/y)]/[√(1/y+2) -√(1/y)]
=lim(y->0)[√(y+1) -1]/[√(2y+1) -1] (0/0)
=lim(y->0)√(2y+1)/[2√(y+1)]
=1/2
lim(n->∞)[√(n+1) -√n]/[√(n+2) -√n] =1/2

(2)
lim(n->∞)5x/[(1+x)^(1/3) -(1-x)^(1/3)]
x->0

(1+x)^(1/3) ~ 1+(1/3)x
(1-x)^(1/3) ~ 1-(1/3)x
(1+x)^(1/3)-(1-x)^(1/3) ~ (2/3)x

lim(x->∞)5x/[(1+x)^(1/3) -(1-x)^(1/3)]
=lim(x->∞)5x/[(2/3)x]
=15/2

(3)
lim(x->4)[√(1+2x)-3]/[√x -2] (0/0)
=lim(x->4) 2√x/√(1+2x)
=4/3

(4)
lim(x->5)[√(x-1)-√(x^2 -9)]
=lim(x->5) (8+x-x^2)/ [√(x-1)+√(x^2 -9)]
=(8+5-25)/(2+4)
=-2

(5)
lim(x->1)[√(5x-4)-√x]/(x-1)
=lim(x->1)(4x-4)/{(x-1)[√(5x-4)+√x]}
=lim(x->1)4/[√(5x-4)+√x]
=4/(1+1)
=2