几道求极限的数学题,1.lim n趋向无穷 （1-1/2的方）*(1-1/3的方)*……*(1-1/n的方)2.lim x趋向2 (x的方-4)/(x的的方-3x+2)3.lim x趋向0+0 (根号下1-cosx)/sinx4.lim x趋向正无穷 【(根号下x+2)-根号3】/x-1
2019-04-15
几道求极限的数学题,
1.lim n趋向无穷 （1-1/2的方）*(1-1/3的方)*……*(1-1/n的方)
2.lim x趋向2 (x的方-4)/(x的的方-3x+2)
3.lim x趋向0+0 (根号下1-cosx)/sinx
4.lim x趋向正无穷 【(根号下x+2)-根号3】/x-1
优质解答
1.n→∞,lim(1-1/2^2)(1-1/3^2)……(1-1/n^2)
=lim(1-1/2)(1+1/2)(1-1/3)(1+1/3)……(1-1/n)(1+1/n)
=lim(1/2)(3/2)(2/3)(4/3)……(n-1/n)(n+1/n)
=lim(1/2)(n+1)/n
=1/2
2.x→2,lim(x^2-4)/(x^2-3x+2)
=lim(x+2)(x-2)/(x-2)(x-1)
=lim(x+2)/(x-1)
=4
3.x→0+0,lim√(1-cosx)/sinx,此时sin(x/n)>0,(n∈N+)
=lim(√2*sin^2(x/2))/(2sin(x/2)cos(x/2))
=(√2/2)lim(sin(x/2))/(cos(x/2))
=0
4.x→+∞,lim(√(x+2)-√3)/(x-1)
=lim(x+2-3)/(x-1)(√(x+2)+√3)
=lim1/(√(x+2)+√3)
=0
有不懂欢迎追问
1.n→∞,lim(1-1/2^2)(1-1/3^2)……(1-1/n^2)
=lim(1-1/2)(1+1/2)(1-1/3)(1+1/3)……(1-1/n)(1+1/n)
=lim(1/2)(3/2)(2/3)(4/3)……(n-1/n)(n+1/n)
=lim(1/2)(n+1)/n
=1/2
2.x→2,lim(x^2-4)/(x^2-3x+2)
=lim(x+2)(x-2)/(x-2)(x-1)
=lim(x+2)/(x-1)
=4
3.x→0+0,lim√(1-cosx)/sinx,此时sin(x/n)>0,(n∈N+)
=lim(√2*sin^2(x/2))/(2sin(x/2)cos(x/2))
=(√2/2)lim(sin(x/2))/(cos(x/2))
=0
4.x→+∞,lim(√(x+2)-√3)/(x-1)
=lim(x+2-3)/(x-1)(√(x+2)+√3)
=lim1/(√(x+2)+√3)
=0
有不懂欢迎追问