求下列极限1. lim(x->0) sin2x/根号x+3-根号3 2. lim(x->∞) (3x-2/3x+1)^2

1.l im(x->0) sin2x/根号x+3-根号3
= lim(x->0) sin2x(√(x+3) +√3) / x
= 4√3
2.lim(x->∞) (3x-2/3x+1)^2x
= lim(x->∞) (1-3/(3x+1)^2x
= lim(x->∞) (1-1/(x+1/3)^2x
=e^(-2)
3.lim(x->0) In(1+x^2)/2xsinx
= lim(x->0)2x/(1+x^2)(2sinx +2xcosx)
= lim(x->0)2/(1+x^2)(2sinx/x+2cosx)
=1/2
4.lim(x->0) In^2(1+x)(根号1+x -1)/arctanx(1-cosx/2)
lim(x->0) In^2(1+x)/(1-cosx/2)
= lim(x->0) 2ln(1+x)/(1+x)(1/2sin(x/2))
= lim(x->0)4/(1+x)[sin(x/2)+1/2 (1+x)cos(x/2)]
= 8
lim(x->0) (根号1+x -1)/arctanx
= lim(x->0)1/2(1+x^2)√(1+x)
= 0.5
lim(x->0) In^2(1+x)(根号1+x -1)/arctanx(1-cosx/2) = 4
5.lim(n->∞) (1/根号n^2+1 + 1/根号n^2+2 +.+1/根号n^2+n)
1/根号n^2+1 + 1/根号n^2+2 +.+1/根号n^2+n
> n* (1/√n^2) = 1
1/根号n^2+1 + 1/根号n^2+2 +.+1/根号n^2+n
< n* [1/√(n^2+n)] = 1/√(1+1/n)
lim (n->∞) 1/√(1+1/n) = 1
lim(n->∞) (1/根号n^2+1 + 1/根号n^2+2 +.+1/根号n^2+n) = 1